I'd recommend a St Croix rod. They're excellent quality and very reasonably priced. An Avid A804.4 will serve you very well ($210) and you could use it for the rest of your life. If you want something a little cheaper, their new Reign series is pretty nice too. You can either get an R804.4 (4 piece, $150) or R804.2 (2 piece, $130). If you're planning on traveling with the rod at all the 4 piece is worth it. You could even save yourself some dough and get a Premier series ($80-100), but I think the Reign is worth the extra money.
For a reel, just find something inexpensive for about $40. It's just going to be a glorified line holder so there's no need to spend much money on it. Just find something you like the looks of that balances the rod well. St Croix sells a Premier CLA 456 for $40 that would work fine. I have to respectfully disagree with jlsimithii, I think $100 is a horrible waste of money when you have a limited budget since the rod and line are the most important things for fishing like you'll be doing in the Catskills (grew up in the foothills of them).
Budget yourself $60 to get a good quality line. Don't skimp on line. Any of the major brands really work fine. I have a preference for Scientific Anglers for my freshwater lines and don't think you can go wrong with those.
So with the cheaper Reign rod, you're looking at $130 + $40 + $60 = $230. This leaves you some of your budget for the misc stuff like flies, leaders, tippet, hemostat, line clippers, etc.
FYI, you could go the Orvis route. Their Clearwater and Streamline series are good entry level rods in your price range.
My passion for catching fish is eclipsed only by the fish's passion not to be caught.