The approach I took, and it may not work for everyone, was to start off by looking at the boats that come up frequently on Craigslist. For me in my price range, those were Hunters, Catalinas, Pearsons, Ericsons, and Irwins. Not that I excluded ANY boat, but as a threshold issue, I learned more about those. I went to SailboatData.com and learned about the beam and the draft. I went to SailingTexas.com and got a feel for what the cockpit and cabins looked like. Then I tried to envision myself and my family using that boat. For example, some boats have a galley that runs the length of the starboard side and a dinette on the port side, with no other seating in the "lounge" area. I knew that wouldn't work for us, because we frequently sail with more than just us, and if it rains, we wouldn't all be able to hide in the cabin. I also looked at the cockpits and tried to envision my family and friends there. Where would kids sit? Are there moving parts (e.g., traveler) that might hit them or pinch fingers? How safe will they be if they stand on the seat to get a better view? I also decided that I wanted a wheel for steering; others love tillers (and I understand why), but for my purposes a wheel was very important. I also knew I wanted a roller furler for the headsail. Issues like those allowed me to rule out some boats, and to push others higher on my list.
Once I had a good feel for the boats that I'd likely run into, I started looking more earnestly at Craigslist and Yachtworld. My budget was a lot lower than yours, so I didn't really focus on YW too much - brokers add to the seller's cost, and I was looking for value-priced boats. Incidentally, I DID wind up buying a boat that was listed on YW, but it was more by accident. Anyway, the next thing I would do is to start examining all of the pictures that were in the Craigslist ad. If I saw stained woodwork near where the chainplates attached, the boat was typically ruled out (unless I suspected that the damage wasn't that bad due to other pictures). If I saw lots of places where the hull had been repaired, the boat was ruled out. If the boat seemed to have been poorly maintained, it was ruled out. If there was a clear line at the keel/hull joint, and especially if there were other signs of trauma in that area, the boat was ruled out. By contrast, small stains on the woodwork under windows didn't really bother me - that means the window needs to be rebedded, and I could handle that (I typically checked to see if the core in that area was rotted when aboard).
Another important thing to learn is that there are a lot of really bad salespeople out there, and they have NO CLUE how to put up a Craigslist ad. They'll have a BEAUTIFUL Pearson 323 in immaculate condition, and the subject line for the ad will be "Pearson 323" and inside the ad will say something like "1980 Pearson Sloop, desal (sic). Looks great, runs great. $8K. Calls only." And that's the entirety of the ad. You want to learn how to use Craigslist's search feature to find those ads, because most people are searching for "sailboat" or "sail boat" (which will return different results if you're on CL), and will never see the Pearson ad, which is actually good for you, because the seller will likely be frustrated that he hasn't gotten any calls. So, when you find that ad and call him, he'll be ready to make a deal.
Once you decide to go look at a boat, Faster's comments above are spot-on. Granted, someone can clean up a boat and give you the impression that it's well maintained, but from what I've seen, most of the sellers who are too lazy to maintain the boat are also too lazy to clean it up when it comes time to sell. There are a few self-inspection guides online that give you a good list of the things to look for during your walk-through. Don't be afraid to push buttons, turn stuff on/off, flush the head, hoist the sails, etc. If the seller is honest, he/she won't mind at all. I'd plan on being there at least an hour, and probably more like two in that initial visit. Also, take LOTS of pictures, even of stuff you don't think you'll want pictures of, and from strange angles. Take pictures of the mast, and the boom, and both together. Take pictures of the cabin from the companionway, and from the V-berth, the head (get the brand of head in the picture if you can), the engine and engine compartment (if it's an inboard, or the engine bracket and how it is attached, inside the cabin, if it's an outboard), and both sides of the chainplates. Take pictures of the floor, and the cabin ceiling. The cockpit, the transom, the sides, the mast step, the anchor area, etc. Get a shot of the Hull Identification Number and/or occupancy plate. Of all things be SURE to get pictures of ALL of the BAD things you see, and from six different angles.
Once you think you've found an interesting boat, load the pictures of that boat into a Photobucket account (or any other file sharing site) and post the link here. There are a LOT of good people here who can help scrutinize the pics and give you feedback. And, as I said before, if you want someone to go with you on a few look/sees, ask. As Faster said, for some people, visiting sailboats is a wonderful way to spend a Saturday!