If you're interested in serious mountain climbing, your second priority (behind getting in great shape) must be obtaining proper equipment and gear. While state-of-the-art mountaineering gear is not always necessary, you must be sure your gear meets the safety standards endorsed by all climbers and climbing associations.
Rock Climbing
Rock climbing gear is far different than hiking gear. For rock climbing, at a minimum, you'll need rock shoes, a harness, a belay device, carabiners, a chalk bag and chalk, and a huge amount of rope. If you plan to do indoor climbing only, you may not need the rope right away. However, as you graduate to more challenging ascents and routes, you'll find that you want to head outdoors.
Hiking Gear
For standard hiking trips, the gear may not be as necessary. For example, a backpacker will probably want all of the following: backpack, sleeping bag, sleeping pad, shelter, water carriers, headlamp, stove, first aid kit, and trekking poles. However, unlike rock climbing where a harness is ESSENTIAL, a stove isn't entirely essential for a backpacker. Some backpackers survive on cold food for extended trips.
Mountaineering
Mountaineering gear is similar to rock climbing gear, but more extensive. In addition to the rock climbing gear, you'll want waterproof and breathable tops and bottoms, heavy mittens and tight-fitting gloves, mountaineering boots, cramp-ons, a helmet, a GPS unit, and an ice axe. Surviving at high altitude on snow and ice requires high-end gear and top-notch training. Do not attempt a mountaineering climb without experience and/or a guide.
