In archery, it's easy to spend a mint getting top of the line gear.
The newest, bestest, fastest, whizz-bang stuff comes out at the ATA show every January.
Compound bows range in price from about $100- 3500, bare bow.
Recurves start at about $130, and go up from there.
Diamond Razor Edge, PSE Stinger, and Bear Apprentice 2 are all great bows for getting started in the $300 range. Not a cadillac, but not a chevette either.
You'll still need arrows and a release to get shooting.
Budget about $500 for a full on start up package.
Tru-fire Hurricane Extreme is an excellent budget release.
I personally shoot a Carter "Like Mike".
Difference? Ultra crisp trigger break, adjustable sear, and silky smooth break.
As for recurves, you have 2 routes you can take. Olympic style, or traditional style.
If you were wooed by Hunger Games, traditional style is what you're looking for.
Samick makes a great starter bow in the Sage model.
It's a takedown style, meaning you can detach the limbs for easier storage and transport.
It fits in a 24" duffel bag.
You can shoot off the shelf if you like, or you can use a rest, or rest and plunger combination, for optimal arrow flight.
Going this route, you'll be learning to shoot instinctively, IE gauging the target point of aim, by gapping- looking down the arrow and estimating how much holdover to hit where you want it to.
Its not easy, especially starting up, but it can be learned.
My best recommendation is to find people in your area that shoot the style you want to learn, and talk with them.
Take a lesson or two.
Fling arrows at targets all you can.
Then go practice some more!
Have fun with it!
Shoot warm shook up cans of soda at 20 yards! Talk about awesome reactive targets!