What's the best way to migrate my files to my new computer?

There is no program that can effectively sort out the useful stuff from the junk. You have to do that manually.
Get all the stuff that you know you need, and transfer it. And then archive all of the rest until you’re ready to go through and sort it properly.

I agree with @Nullo; there is no substitute for human discernment.

I would look at it from the level of apps first, then files. What apps do you have and use that are not on the new Mac? Move or reload those first. What you’re then going to want to move is music, photos, Office docs – word, excel, ppt, etc. and anything you’ve opened and saved from the internet that you want to keep. Clean out and sort what’s in the All Images and the All Documents folders on your hard drive; also go through what’s saved under your name.

I always intend to take the vow of better file management when I get a new computer, and fail miserably.

@kfingerman I don’t know if I’m reading you properly but I think you might be concerned about little programs that were installed in your system here and there things that worked their way into your preferences – to sort that out would be a pain right? In general you just want to start out with a clean ID right?

I had a similar thought- I had previously moved everything over manually before migration assistant existed and it had worked out ok but I also wanted to make sure I wasn’t forgetting to move anything on my old PC. What I did was migrate my old user using migration assistant to new computer while setting up a new user on the new computer. I decided to move over my applications manually because I didn’t want to replace any of the newer applications that might be on my new computer, and I didn’t really want to bring over any of the old applications that I don’t use anymore.

Then I transfered the data from my old user on that computer to the new user and then deleted the old user. I’m not sure if this is the best idea exactly but it’s worked out ok and assured me of a little cleanliness. If you do decide to do this I would be comfortable with changing permissions though- because you will run into permissions problems when you try to transfer the data between users- you might have to go into some of folders and manually change the permissions so your new user can access the data. That’s kind of a pain but it didn’t take long.