Shoes
A good place to start are basic climbing shoes; you will need them no matter what type of climbing you will be doing. Climbing shoes are specially designed with a sticky rubber sole to assist in grip needed to climb. There are many manufacturers and many types and styles of shoes to chose from. Finding a pair that fits your style of climbing is as important as finding a shoe that fits your foot.
Harness
If you will be participating in climbing other than bouldering you will need a climbing harness. In recent years several manufacturers have begun to produce harnesses specifically designed for the type of climbing you will be doing. You will find lightweight and simple harnesses geared towards gym and sport climbing as well as harnesses meant specifically for all day comfort on multi-pitch aid routes. Take the time to try on several and get some advice as to the proper fit and procedure for wearing a harness, as your life will depend on it.
Additional Equipment
If you have decided that climbing is for you, you will need additional equipment as you progress through the sport. This equipment is varied by type and style of climbing, but a few basic pieces are invaluable. A good climbing rope is a great place to start. When climbing with others I often prefer to use my own rope, that way I know just where it has been and how it has been used. You will eventually want to begin adding active and passive protection, including spring-loaded camming devices (SLCDs), nuts, offsets, quick-draws and runners. With so many different pieces of equipment available, it will take time to learn the proper and safe use of each. Classes are often available through your local climbing gym and can be very informative. Be safe and enjoy the climb.