I don't have experience with the 5200, but from what videos I've watched, it seems like they have a problem with shitty controllers. So check for that on listings.
As for the console... well, it just depends on what you want. Are there particular games you want to play that are only available on the 5200? If so, are they affordable for you personally? Are they worth the price of entry (price of console)? And, since I'm not familiar with 5200s, does the console tend to have any hardware problems etc. that it's infamous for or prone to? Also, keep in mind that Ataris are very, very easily emulated too. Obviously most of us like to have physical merch, but that's another thing to consider as far as price of entry goes.
Personally, unless you have some specific games you really want a 5200 for, I might pass on it. While it's true that you'd save money down the road by buying a system that plays 5200 and 2600 games, you may also come to realize that you do or don't particularly like that console/it's games as you collect more of them. Hell, for $65 buy it now on Ebay you can get a 2600 jr and 10 games (approximately 1/3rd of the maximum you're willing to spend right now). I also found this 2 port model of the 5200 for a much cheaper buy it now price than you were looking at.
http://www.ebay.com/itm/Atari-5200-2-port-model-with-ac-adapter-and-3-games-/282353327838?hash=item41bd91dede:g:kr8AAOSwo4pYmekOHell, here's two of 'em haha.
http://www.ebay.com/itm/2-ATARI-5200-4-PORT-SYSTEMS-TESTED-WORKING-4-CONTROLLERS-8-GAMES-/332116781716?hash=item4d53b3ea94:g:haIAAOSw44BYd~QaSo in a sense, i guess since 5200s are around the same price, you could go with one of those if you really want. But remember the above questions, and also look into the controller situation. I don't know how popular garage sales are around your area, but I guess it's always possible you could find a system at a pawn shop or something for cheaper, too (probably not a 5200, but a 2600 maybe).
In any case, I'd lower my budget. Looks like you should be able to get started for under $100 easily.
Personally, I'm not an Atari guy, so maybe others will chime in. I have a 2600 and will pick up a game for a buck now and then (Atari game art is among my favorite kind), but I usually just emulate since it's more convenient.
*EDIT: And obviously, call/check out your local (nor not so local haha) retro shops! Even if they don't have the system, you can see if they have games (that way you can buy locally/semi-locally, Ebay prices can suck it lol). It's good to have a source instead of having to buy off the internet all the time. My not-so-local Game X-Change mostly had 2600 stuff, but there was a 5200 game or two popped in there as well, I think. And maybe an Intellivison and/or pong system, I'm gonna have to check it out next time...
