Ok, the 2002-03 "bugeye" and the 2004-05 "blobeye" WRX's are very similar mechanically. They will have the 2.0L EJ20T motor.
-Look for evidence of previous mods. This is VERY IMPORTANT with these cars. These cars do not like traditional cold air intakes (they will ruin lean), atmospheric vent blow-off valves (they will dump fuel), manual boost controllers (will screw with the fuel map), and other typical mods that other cars like Hondas and DSM's, work with. Look at the exhaust system and check if there is new hardware or aftermarket parts, which is evidence that someone was messing with the car.
Smell for coolant too under the hood. The very first WRX I looked at way back in 2005 once had a different turbo in it. I could smell coolant when opening the hood, and sure enough, there was a new coolant feed line on an old turbo.
-The transmissions have a bad reputation for being "glass transmissions", or that they break easily. I had a 2002 wagon that I drove with "Stage 2" mods and raced it occasionally and I never had an issue. Just don't drive like a meathead and you will be fine. Put good fluid in there and you shouldn't have an issue. Do not use Mobil 1 gear oil though, as their additives are not compatible with the synchros used in the stock 5-speed and will cause binding. I learned that the hard way.
-Other driveline components, like the rear differential, should be stout enough to take abuse. Regular WRX's have the R160 limited slip rear diff. Only issue I had with my 2002 is that at approximately 62mph it would whine. It was due to weird harmonics in the housing, and I believe the later ones don't have this issue.
-On your test drive, try to load up the rear a bit and shift aggressively a few times. If the rear diff clunks, you will need to get rear subframe bushings as well as rear diff dogbone mount bushings. They are easy to do, and stiffen up the rear of the car quite a bit. I recommend the Kartboy stuff. They are affordable and proven.
-They make all sorts of suspension bits for these to outfit your car to whatever goals you want to achieve. You want to do autox? They make everything you could ever want and then some. Lots of great companies make springs, sway bars, braces, and other things. If you get a wagon, invest in a bigger rear sway bar and possibly a rear strut tower bar. My wagon had a great setup: I had the STI "pink" wagon springs (bought right from the Subaru dealer), Tokico D-Spec adjustable struts, a 2002 WRX sedan 20mm rear sway bar, and an eBay STI-replica rear strut tower bar. The car handled incredibly well.
-Look for rust in the front control arms. They have a tendency to rust there. Wagon ones and sedan ones are different. Check the rocker panels too. I've seen them rot there as well.
-I'd install some gauges. WRX's don't have much for gauges, and with how finicky they can be, you should have them. I had an EGT, oil pressure, and boost gauge in an ATI clock pod on the dash. I'd spring for some high quality gauges. I had Prosport gauges in my car, and they were a bit flaky, especially the EGT.
-For the brakes, they are pretty standard with the integral parking brake drum-style rear discs. There are lots of upgrades out there, but I had a very simple and effective setup in mine. Stock calipers, Hawk HPS pads, regular "blank" rotors, stainless steel braided brake hoses, and ATE Super Blue fluid. To this day, they were the best brakes I have had in any car.
-These cars are easily tuned. This is a great thing, but can also be a bad thing too. If the car has a bad tune or a tune for mods the car doesn't have, it could lead to issues. I'd invest in a Tactrix cable or make friends with someone who has one. It's cheaper than the popular Cobb Accessport and you can do more with it. They are great for data logging too. I'd do an off-the-shelf "stage 1" tune to start because it cleans up the fuel map and is probably safer than the stock map!
-I'm pretty sure that in either 2004 or 2005, Subaru switched to the non-catted exhaust up-pipe. If it has a cat on it, swap it for a non-catted one. Just like Nissan Sentra SE-R's from the early 2000's, the cat material can break up and back feed into the engine (and turbo) and cause issues. And if you replace it, don't cheap out. The cheapo flex joint ones can cave in and fall apart too!!! You will have to tune out the factory EGT light, which you can do with a Tactrix cable or Accessport.
-These are very sensitive to whatever oil you use. Don't go over the oil change intervals with these, and run a reputable full synthetic oil. I ran Shell Rotella 5w40 full synthetic meant for diesels. I ran at elevated boost levels for over 60k miles with no issues.
-Interiors for the 04-05 cars are generally nicer than the 02-03 cars. Not much to really look out for, and I suggest upgrading the stereo. The stock receiver is TERRIBLE, and it's a very easy upgrade.
-If you are buying a manual-equipped car, invest in the Kartboy short shifter kit and bushings. It makes a world of difference.
Did I miss anything? Let me know if you have any other questions. 
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