Bouldering: Climbing Shoes, Chalk bag, Comfortable clothes (e.g. knee-covering pants, sports shirt)
Top Roping: Same as above, Climbing Harness, Belay device.
Lead Climbing: Same as above answers, Rope.
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That really depends on what style of climbing you are doing.
For trad, sport, aid, and ice you will need a harness, rope, carabiners, helmet, protection (specialized gear like quickdraws or cams), and some specialized shoe for the style.
For bouldering you need shoes and a crashpad.
For deepwater you just need shoes (maybe).
Chalk is also nice.
don't just go buy gear and try it out though. Go with someone who knows what their doing, learn the sport, and use their gear for a bit.
The most important piece of safety gear one needs for rock climbing is a helmet in order to protect one's head in case of a fall. One needs a safety harness and rope to as well as gloves.