To be honest, snprbob86's criteria are a lot more robust than mine and probably work for more people, but since you asked, I'll share.

My initial policy was to look for old games that had been so successful that they were re-released in new bargain versions. For example, my senior year I bought a jewel-case pack of Fallout and Fallout II. Basically, my thought process was: If it's so good that they're still pressing new CDs, I'll give it a try. Of course, I still had to apply my "repeat simple action, increment arbitrary number" filter, because Diablo and Diablo II were released in a similar way.

I still generally only play older games. It makes it easier to tell the good from the bad (plenty of reviews) and it saves me money.
EDIT: Another advantage of old games is that when you buy a stinker (and you will, no matter how picky you are, occasionally buy a game that turns out to be no fun at all), you don't feel the same pressure to finish the game anyway that you might feel if you had dropped $40+ on a brand new title.

After college I removed the Windows partition from my computer and stopped dual-booting, so ever since then I've only played games that I can run on Linux. I prefer games that actually have Linux versions, but I run plenty of stuff on dosbox and even some stuff on wine. I played a lot of Neverwinter Nights: it has an amazing community that still produces new modules even today. I'm also a big fan of Descent (all three versions). For a long time, my main criteria was only to pick games that really resonated with my imagination (and, of course, apply my filter to avoid exercises in "incrementing the number"). For example, I'm a sucker for steampunk so I picked up Arcanum, and it was excellent. I couldn't get it to run on Wine, but I wanted to play it so badly that I set up Windows on VirtualBox so I could play it.

I like first-person shooters, and find that they're not as prone to the "incrementing" problem; ditto for strategy games. Story-driven games are generally immune to that problem, but can fall prey to a different problem: excessive linearity (i.e. you might as well just read a book). About four years ago I went on a kick where I re-played a bunch of old Sierra adventure games on ScummVM. I think that there are some indie developers out there now making similar games.

We had our first kid three and a half years ago, and as a result I've had to change my gaming habits yet again (work also got way busier at about the same time, so it's not just the kids). I now prefer games that I can play for short spurts. I was playing QuakeLive for a while, but got bored with it eventually (and all the trash talk didn't help). I re-discovered nethack (I still suck at it, but that doesn't make it any less fun), and you can't go wrong with minecraft. I played Steambirds on Android, and that was a lot of fun but didn't last long. Also on Android I've found a couple of pretty good games that remind me of "Incredible Machine:" "Clever Contraption" and "Electric Box."

EDIT: I forgot to mention Wii! There are a lot of Wii games that are just plain crap, but there are a few real gems. Wii Sports Resort is simplistic but still lots of fun, and fits great with my "short spurts" approach. It's also something my wife and I can enjoy together with our eldest. The only really "serious" game I've enjoyed on the Wii was Red Steel II (don't play the first one--it was awful!).