ANYONE in reasonably fit condition can begin to climb. However, as in other sports, the better shape you're in, the better you'll be at the sport and the more you'll enjoy it.
The single most important thing is to shed any excess weight. Your strength-to-weight ratio is one key to successful climbing, and getting rid of a spare tire around your middle probably is the fastest and easiest way to improve this ratio, and with it your ability to climb difficult routes.
But strength is important, too, in your legs and upper body, particularly hands and arms. Many climbers run or jog. Others work out on Nautilus or other weight equipment.
The object is not to end up looking like a weight lifter, and few good climbers do. Rather, you should be able to raise your own weight easily. For the upper body, exercises that emphasize reaching upward, such as chin-ups or the equivalent, are best.
One inexpensive way to work on your forearms and strengthen your hands is to take an old broom handle, drill a hole in the center and attach a gallon milk container to it with heavy string. Fill the container with water and raise and lower it by winding and unwinding the string around the broom handle.