You should be able to get a setup within your budget that will work just fine. Both Scientific Anglers and Cortland make complete outfits that come with rod, reel, flyline and backing. I think the SA version even comes with a tapered leader and a flybox too. The prices have probably increased since I last recomended one of these to a friend about 6 or 8 years ago. As I recall they were about 160 to $180 at that time.
I'd also recomend a 9ft #5 rod as a an all-around trout rod. A #6 would be my second choice.
Don't let anyone tell you that you need to spend a lot of money to get a decent outfit to get started flyfishing for trout. It's nonsense! There is a lot of junk out there though so be careful. Places like Walmart are notorious for poor-quality equipment that will be more costly than purchasing something decent. The SA and Cortland outfits will work fine and will allow you to spend more on things like flies, fly boxes, leaders and tippet and waders. You can buy nicer (and more expensive) gear later when (and if) you you find yourself hooked.