I’ll assume you’re on Windows, because Mac already has the awesome iLife editor that comes with most of their systems, and most Linux people know where to look for binaries. Correct me if I’m wrong though.
I’ve never used any free video editing programs, as my schooling taught us the tools that are used in the industry, and very few open source solutions have the market penetration/abilities as the pay one. As well, it’s worthwhile to note off the bat that most “legit looking music videos” are not only edited using expensive software (Final Cut Pro or Adobe Premiere Pro), but are shot on a precisely controlled set using expensive lights, high definition cameras, etc. As well, they have the money to pretty much do anything, so if a certain effect they want doesn’t exist they can pay a programmer to make it exist. This isn’t to say you can’t make your own videos of course. It’s just a good fact to consider in setting your goals :)
Here’s a free software I just found. It’s called Wax. It seems to have the basic functionalities you’ll need and you can download even more effects from this person. I’d say it’s a good place to start, be sure to keep a note of anything you don’t like with the software, so that you can make a “shopping list” of features you want if you have to find another free software. I’m not sure if it would edit the audio inside the program – the professional tool I us had it set up to edit the audio in another program. You may have to look for a separate audio editing software.
I’d be interested to see what you come up with :) Good luck.