Warning: Wall of text incoming
What 4th edition did mostly right:
1) Giving weapons an attack value as well as a damage value
This allowed for more depth in choosing one's weapons. Generally speaking, I like it when weapons have a wide swath of statistics to differentiate them from their peers. Also, weapons are only situational in very rare circumstances. Ordinarily, you only have to calculate how a weapon affects your statistics once when you acquire the weapon. You of course have to modify them if you find a better weapon, but it is rarely an involved process. However, in 4th edition there were some weapons that were objectively better than other weapons in a similar category. I would hope that when creating weapons in the future, weapons are created using a sort of budget, where each increase in a statistic uses some of that budget, that way the weapons are closer to side-grades rather than upgrades. Some weapons may be subjectively better depending on one's build or objectives, but where one weapon is objectively better than the others in the same category should be avoided. That being said, I have no problem with say "martial weapons" being better than "simple weapons" by being built with a higher budget. I just want the methodology to be internally consistent.
2) Healing Surges
I liked how 4th edition divided it between how much damage you could take right now in your hitpoints, and had your healing surges for a sort of reserve for how much damage you could take in a day, roughly speaking. I also liked how many of the healing abilities were based off of healing surges. The more robust characters could heal more with the same effect. It made the robust characters feel as if they were more resilient. I also liked how the healers would heal additional damage on top of the character's base healing surge value, showing that healers could greatly aid someone but it wasn't entirely dependent on them. For the same reason, Second Wind was a good, though hopefully rarely used, option for all classes.
3) Creating Fortitude, Reflex and Will Defenses
This just made it so that attackers and defenders used the same methodology across the board. Furthermore, in other editions spells would automatically hit, and it was up to the defender to make their save. This evens the playing field so to speak. It makes it so that players set their defenses, and then it is up to the attackers to try to overcome it. It is my experience that a player is more upset when they fail a crucial saving throw than when an opponent rolls well on an attack against them. Though this observation may be colored by the prominence of the so called "save or die" effects in the previous editions, which have been removed in 4th edition. In my opinion, 5th edition should keep this system.
4) Streamlined Monsters
Reducing many of the monsters and opponents to a state where their statistics could be kept on a single index card greatly helped move combat along in my opinion. In addition, having monsters be worth a certain amount of experience points, and having different encounter levels be constructed by reaching a given budget is a good system as well. The templates and roles for monsters were also a welcomed addition. Templates allowed for easier customization of that one powerful monster, and roles helped to define how to tactically use a monster in a given battle. Having pre-made encounter groups was also quite a boon. I would have liked to see the listings for the monster encounter groups in one place rather than scattered throughout the book, but that's just nitpicking on my part.
What was a step in the right direction, but has more room for improvement:
1) At-Will, Encounter and Daily Powers:
I liked the Powers, but they were poorly executed. A RPG system needs to have both ease of play and depth. Both facets are important, and Powers, done properly, can add just that. What I mean by ease of play is that once your character sheet is nice and finalized, it should be very easy to determine how your character performs a given action. Powers are basically a short hand for a type of action a PC might perform. But the important bit for me, is that how I arrived at this particular collection of numbers could be totally different from someone else of my class and level.
Where Powers failed was by not adding depth to the game.
So here's how one might add depth to the game. I personally liked the tiered sets of powers (At Will, Encounter, Daily). However, I do not like the one and done formula of these powers. What I mean by this is that you can only use a given power once before it is expanded and can no longer be used for the remainder of the encounter/ day. In addition, missing with a daily power quite frankly sucks. Most of the metagame therefore revolved around increasing one's attack bonus as high as possible, which also fed into the concept of a party "nova" where the team would focus on an enemy by using many abilities to increase the party's accuracy, and damage, while reducing the target's defenses. My complaint is not that this is a viable strategy, but it became practically a necessary strategy so to not waste the limited resources that were daily and encounter powers.
I believe that this can avoided in the future by adopting this philosophy: The less frequently a power can be used, the more likely it should be to succeed.
To that end I propose a system similar to one used in MMO's and tabletop games like Iron Heroes. Throughout the course of the battle, a character gains "resource points" which they use to fuel their more powerful abilities. The more powerful the ability, the higher its cost to use. Now battles becomes more strategic. Do you save up your resource points to use your big whallop ability, or do you spend them as quickly as you get them using your middle of the road abilities?
But, with the 4e system, a player didn't have much in the way of real choice in selecting their powers at any given level. When selecting powers in 4e, the question was always "What is the best power at this level?" instead of the far more important "What type of powers do I want to have?"
So get rid of that system, and let players create their own Powers. For ease of discussion, I'll use the terms of At-Will, Encounter and Daily Powers. I will also use a Barbarian class, whose Resource Points are called Rage. At Will powers are the ones that build a character's Rage, which are then used to fuel a character's Encounter and Daily. And because the players can make their own Powers, we have to come up with some sort of framework for building them. So lets say each Power has a "Construction budget" of: 1/2 character level, the character level and 2x character level, for At-Will, Encounter and Daily Powers respectively. Let's say that each different Power has a given Framework, which broadly defines its shtick, which is "free" for the budget. For example, one Framework may be a self heal, while another is a high damage single target melee power, while yet another buffs your character temporarily. Each Class has different Frameworks available to it, though there is some overlap. A Wizard and a Fighter likely have wildly different Frameworks, but a Fighter and a Barbarian are probably similar in many regards.
Then, a player can tailor each of these different Frameworks with Templates that add new characteristics to the Frameworks, such as an increased bonus to hit, more damage, a debuff to the target, etc, each with their own different costs to the "Construction Budget". These Templates may only work with certain Frameworks, and they may even have feat or level requirements. There could even be multiple levels to the Templates, like Accuracy I, and Accuracy II, which gives a +3 or +5 bonus to hit respectively. The more Templates attached to a given power, more powerful it is, but it also requires more Rage to use. Have the Construction Cost of the Power determine how much Rage it takes to use. A character can decide not to use the maximum number of templates on a given power, either because they are saving up for a template that is currently out of their reach, or maybe they just want to keep the resource point cost down, either way it is their choice.
A character may only start play with 2 At-Will powers, 1 Encounter Power and 1 Daily Power, but as they level up, they should gain more Powers, which can be further customized as they see fit. It should be noted that when a Power is gained is irrelevant to its budget or structure. Let the player customize their powers as they fit, regardless of when they were acquired.
So with this system, characters have the simple Powers like 4e did, which simplified and sped up combat, but the Powers all do precisely what the players want them to do. And if you want to keep a "pick up and play" style to the game, the include pre-made Powers in the game, showing how they increase with level. It lets the new players get right into the game, but also allows the veteran players to get some new ideas, and see how everything comes together.
2) Class Features and Roles
This continues my point above. Roles were a good addition to the game. But, I don't particularly care for the fact that a given class only has 1 role. I would like it if a given class had more than 1 role. The character classes in 4e had different builds and therefore played differently, but had the same features and "role" which didn't particularly mesh well in my opinion. A Rageblood Barbarian for instance played much like a Striker, but a Thaneborn Barbarian had many more Leader-like options, but was still considered to be a Striker.
Also, going off my power construction idea above, a character should gain resource points when they do something their role should focus on doing. Defenders gain resource points when attacked, strikers get resource points when they bloody or defeat an enemy, etc. One could also gain resource points for "acting in character" both in meta-game terms for good roleplaying, but also mechanically for a character taking certain actions in combat. A barbarian gains resource points for instance by taking an action to bellow a war cry at his foes, or a rogue gets resource points by attacking from the shadows. Take a look at the Iron Heroes system for more idea on how this can play out in an RPG. In addition, what I would like to see make a return is more static class features of the different classes. I would especially like the ability to pick one's class features in Talent trees like you do in Star Wars: Saga Edition. That is a good system. It also lets a player refine their character's role and how they play. Options are good, there should be more of them at all levels of play.
3) Magic Items
Magic items are quite prominent in this latest version of D&D, which is something I quite like, not that the previous editions were lacking mind you. I also like the inclusion of different item levels, which lets players and DM know what should be accessible for a given level. What I don't like is the encounter and daily powers of the items, something I disliked from the 3.5 days as well. An item either has the effect or it doesn't. None of this one and done stuff. Now don't get me wrong, the spamming of certain abilities can be game breaking. So have certain magic items have their effects powered by a character's resource points. So for example, we've got the Boots of Swift Charging. They have a static effect of allowing a character to move 1 extra square when making a charge attack. The Boots have a secondary effect of giving the character the option of spending some resource points to get some extra squares of movement. More options that can be done frequently throughout the encounter
4) The increased number of feats
Getting a feat every other level is a good step, but I would rather it be a feat at every level. In 4th edition, there were many feats that a given class simply needed to take in order to function properly. The most blatant of this is the weapon and implement expertise feats. According to the Character Optimization forums, those are the first feats you take, and they're not wrong to. This leads to a character having little room to truly customize their character as most of their feats are already going to being useful in combat. More options is never bad.
5) Utility Powers
Utility powers added a nice bit of variation in a character's repitiore of powers to use in combat in 4e. The inclusion of skill based utility powers was also a step in the right direction. Adding more options for solving problems without having to resort to stabbing things in the face is a good move. However, because of the limited nature of utility powers, both in the number of them that there were and in how infrequently they could be used, they could not be used to their full effect. Furthermore, the choice often had to be made between combat effectiveness and out of combat effectiveness. My question is why? The point of 4e was to put all of the characters, regardless of class they are, to be able to contribute successfully to combat. How about a system where a character gains a utility power for each skill that they are trained in, gaining more as they level up. That way, being adept at a given skill has more application that simply having a larger number to add to a d20, it means more options. A separation of combat applicable traits, and the axillary traits would make the game run smoother in my opinion as it gives everyone a way to contribute somehow in the space in between fights.
5a) Powers and Skill Challenges
While we're on the subject of the divide between out of combat mechanics and in combat mechanics, the Skill Challenges in 4e always rubbed me the wrong way. By reducing the skill challenges to a simple execution of pass/fail, you rob the players of creative ways to solve the problem. In addition, powers that should be able to help you, can't without the GM adjusting the rules. For example, I've got a barbarian character, who after a few ill-chosen words at the local watering hole is being chased out of town by an angry mob. Skill Challenge kicks in. In this scenario, my barbarian is going to use his Athletics skill to put some distance between him and his pursuers. Simple right? Well it is simple, but lets say I've also got the Combat Sprint encounter power. This lets me run substantially faster than normal for a quick burst. Surely this would let me evade my pursuers better. Well no. It doesn't. Not by the rules as written anyways. How about the Fleet-footed feat? Nope. How about if I have an overland flight speed? Again no.
Sure the GM can ad-hoc a solution. But what should he do? Add an automatic success? A bonus on the check? Skip the Skill Challenge all together? By stripping Skill Challenges to their bear essentials, you rob the players of unique ways to approach the problem. Heck, in the Skill Challenges, the players have to roll vs a Hard DC using a Skill other than the Primary ones, and then that skill can be used only once. No. That is the wrong way to do things. If a character comes up with a unique and inventive way to solve a problem, you reward that player, not punish them. Have them roll against a lower DC, not a higher one.