You can rock climb with nothing more than a good pair of climbing boots or shoes (i.e. bouldering, scrambling) but if you want to get into technical rock climbing outside, you're going to need at the very least an alpine or sport-climbing harness, some carabiners, a belay device, a helmet, and some runners. Most beginning climbers will not invest in a climbing rope until they've had a few chances to see if they like it. If you climb at a climbing gym, you'll never have to buy your own rope, as the gyms will have set routes and supply the ropes at each climbing station. As for helmets, they're essential if you're climbing outside, but many people will skip using them if they climb at the gym.
You can check out other great free rock climbing info at:
http://hobbies.expertvillage.com/experts/rock-climbing.htm
