This is one of the easier trophies in New Vegas. Lockpicking is one of the biggest things to do in Fallout. I will not be listing all the locations of anything that can be lockpicked since I don't want to write 4 Wikipedia articles.
Anyways you will want to level up your Lockpick skill by spending your points you get by leveling up. I always try to throw 2-3 points on it every level. You should try to get it to at least 75 as soon as you can.
Here are the thresholds for the lockpick skill:
25 - Easy
50 - Average
75 - Hard
100 - Very Hard