I would recommend going to a good reputable archery shop and getting them to set you up. You can certainly buy a new bow but many will have used bows to choose from and can adjust it to fit you. If you buy a used one from just anyone they usually will have a different draw length than you, therefore different length arrows, etc. Then you might have to buy new cams or other pieces and spend a lot more than you thought you were to get it set up for you. I would strongly recommend to see someone who knows what they are doing and get set up right. You're chances of enjoying it are much higher.
You should check out PSE Bows .You can purchase a top of the line bow for $500-600 .Go to a archery shop. You need to know your draw length what draw wt you can draw, the correct arrows to use, a release peep ht set. So much they can help you with. Semper Fi.PS There are many good bows for less then what you want to spend.
Beanareaux nailed it. Gotta get it from a good shop to get it dialed in for you even if it's new. Plus it will teach you how to use it somewhat. In the area I live in, Jim Bowie'sOutfitter's on Perkins Blvd. in Baton Rouge got some honest, un-snobish, advice and choices of bows. Shotting range right there. They could fit it out for you and get it sighted in. They can fit the arrows to you too. We all don't shoot the same weight/ arrow/tip combination. Get a trigger. 8 arrows, 8 target points, 3-3 blade broadheads. Get a target. Find a feild or appropriate shooting place. Grip/thumb/ya-nose. After you get that down all you have to do is sneak up to a big deer within 15 to 40 yards unobsructed.
For a new bow in your price range, take a look at the Mission line of bows. They are made by Mathews but they are more economical.
If you go to a reputable archery shop and get properly fitted, you can probably find a good used Mathews, Hoyt, or Bowtech top of the line bow that is 2-3 years old already set up for $500-$800 depending on the accessories and age of the bow.
I have one for sale if you wanna know more send me a message. Jeremy@synergyelec.com