I am thinking about getting some video editing software and I was wondering if Premier Pro or Final Cut Pro is better in terms of ease of use, compatibility with other audio and video editing programs, and how powerful they are.
In my experience both Premiere and FCP will do pretty much the same thing. They have similar lay outs and, I would say, are as easy to use as each other. I first used Premiere when I had a PC but moved to FCP when I went to university and got a mac. I find it a lot easier to edit with FCP and the transition to other programs are very simple and effective (file > send to > Color/Compressor).
If you have a mac I would suggest buying the final cut pro studio. It comes with Motion, Soundtrack Pro, DVD Studio Pro, Color and Compressor. Everything you need (pretty much except Adobe After Effects). And if you have a PC then go with Premiere. That's my advice.
I would say, after having used both for a number of years each, that Chard was right up until recently. If you are on a PC then it is a no brainer - Premiere is the way to go. If you are on a Mac it is much more complicated with the new release. Apple has stopped selling FCP Suite 3. Now we are all just hoping they increase the functionality of FCPX. As of right now it is pretty useless without external monitor support, unless you just edit on a laptop. If you need to make a purchase soon, I'd go with an Adobe Suite that bundles Premiere with some other useful software like After Effects and Photoshop.