I think the most important step is to read through the adventure a few times and look up words or references that you are not sure of. We tried to capture everything you need about the setting, but LFR does not repeat rules which are in PHB.
WATE1-1 effectively has 3 major encounters and one minor encounter. The investigation is complex because in a very large city, to find a stolen heirloom, you probably will look for the "fence" who bought it to resell, which leads you to the thief, which leads you to the actual buyer. We tried to design the "scenes" in the Skill Challenge to be fun, role playing. I recommend that you not announce you are going to be doing a skill challenge, but instead try to push a few clues or leads to the players, and let then follow up naturally. If they make a detour, allow it, but swap names and places and try to steer back towards the final destination. Usually when the players role play a scene, I slip in a "that sounds like a diplomacy skill, how about a roll?" or whatever it is. Usually for each scene there is more than one way to gain a success although often only one is needed. Don't get too fixated on the exact number of successes...and if they get to the final destination and don't have the exact number of successes, just ignore that.
We put in the minor combat (dock ward with two young lovers) to IN PART break up the skill challenge, thinking that some players might be getting bored. Also, it shows that the dock ward can be tough for citizens (not so much for adventurers) and is a seed for later contacts. More role playing than combat. Be sure that the players/PCs were fairly, clearly warned in the beginning about the legal system and they should really avoid killing people within the City limits. Game mechanic wise, knocking creatures unconscious works just as well.
The combat with the gang of thieves can be hard if the PCs get stuck on the stairs down (due to the thieves being warned by alarms). Remember two factors: the thieves don't want to kill inside of Waterdeep either (they are thieves, not murderers) and two they would like to get away.
The combat encounter with the gnomes is more about catching the gnomes than beating the gnomes to a pulp. So use all the ingenuity you can with the gnomes to hide and sneak--their preference is to get away, not to fight (with the exception of the guard drake). Many players have found this encounter to be quite fun. The gnomes will surrender when it is obvious that they are caught. They were paid to obtain an item, not to die for their customer.
Make it fun and try to keep the flow moving along. If you have questions when you read it, let me know.