If you thought the bass fishing bait monkey was bad...
There is some good advice above regarding the line and rod being most important 90% of the time. The types of water you fish and the species you're fishing for will have some impact on what weight, but generally, a 5 or 6 wt will cover most trout fishing bases. You can fish dries with them, nymph, and throw smaller streamers reasonably well.
Here's where I suggest spending a bit more money and why:
The thing with spending money on fly gear... bad gear can make casting a frustrating nightmare and totally turn you off to the sport. My dad bought me a cheap Cabelas 5wt combo with cheap line that I almost immediately had to replace because casting was less than user friendly. Also, good gear will last you YEARS so long as you don't beat up on it. Consider these things one-time investments for years of enjoyment.
I suggest expanding your budget a bit, even if it means fewer beers, you'll thank me later. The Fenwick Aetos fly rods generally sell under $200 and can be had on the inter webs for way less (I paid $100 or so for my 4wt). It's action is fast, but manageable for a beginner. Redington is a company now owned by Sage that also makes a lot of high quality products intended to not break the bank for the new fly angler. Still, I would suggest spending the equivalent of $150-$200 on your rod.
Line is next. Getting a floating front tapered line will cover most of your bases. For most trout fishing, I really like Rio Gold, but Scientific Anglers, Sage, Orvis, etc all make quality lines. Some may sell for $70-100, but you can catch them on sale. I don't think you need anything incredible to start, but I would suggest spending at least $50 on line.
You can learn to tie your own leaders once you get comfortable casting and begin to understand specific purposes, knots, etc a lot better and they're a great way to save money. However, at first, I would suggest buying tapered leaders to save you frustration. I will also suggest using tippet rings for just about any application that isn't dry flies because they'll save your leaders. I would also suggest picking up mono tippet in 4x and 5x to cover most applications.
You can learn on a $50 reel and be totally ok.
I will also very strongly suggest casting lessons. Sometimes fish commissions do them for free/cheap. Other times you can get them at fly shops/ Orvis stores. You're going to wan to practice at first. A lot.
So...
if you can expand your budget to $300 or so, you'll be in great shape and won't be in a crazy rush to upgrade everything by your second season.