Rock Climbing Shoes are the first and most important piece of equipment you can purchase to improve your rock climbing performance.
Be careful not to buy shoes that are too tight as you will concentrate on your pain more than climbing and have much less fun. When climbing shoes stretch you can always compensate by adding socks.
Rock climbing shoes should fit tight around your feet. A good-fitting climbing shoe is going to be very snug, especially if you're buying a sport shoe. Make sure your toes are cramming into the toe box evenly. If the toe box feels really tight in one spot and not so tight in another, compare the fit with a different manufacturer's shoes, and your toes should not wiggle.
Every shoe company uses different lasts (hand-carved feet that the shoe is built around) that have little or no correlation to each other so try on a variety of brands and a variety of sizes. Some companies expect that you'll wear a climbing shoe that is one size less than your street shoe size, and others make the adjustment in the factory so your climbing shoe and street shoe sizes are the same. Unlined leather climbing shoes stretch about 1/2 a size with wear, so buy them snug. Lined leather shoes don't stretch as much as they will mould to your foot after a few outings. Synthetic shoes won't stretch at all, so the fit you have in the shop is the one you'll have to live with. Lace the shoes tightly. The eyes of the lacing should not touch - if the eyes of the lacing touch when the shoes are new, they'll be floppy when the shoes are broken in.
(Be aware that after 15-20 pitches there will be some amount of stretch. Consider fit, construction and fabric at the time of buying)
Type of Shoes
There are many types of rock shoe on the market and some are very specific. So you may need more then one pair. And you may need to adjust the fit.
Traditional climbing shoes are for long climbs, including cracks, face, chimneys, and slabs. They generally have a supportive last, the fit is looser than other climbing shoes, and remain comfortable after many hours of climbing. These are recommended for beginners to the sport who plan to spend time climbing outdoors on real rock as they have a lining that protects the feet and is easier on the leg muscles, especially the calves.
Sport climbing shoes are for mostly face climbing, limited crack climbing, and climbing walls. They are mostly slip-lasted and a tighter fit, more flexible and have more rounded edges. If you plan on spending time indoors, beginner or not, you should go with sport shoes.
Performance shoes are generally for overhanging sport routes, bouldering, and competition - not for beginners. Climbing in performance shoes will strengthen and harden your feet up. This type of climbing shoes have a zip or elastic closure. They are fast to get into, but will hurt your feet edging and standing around if you are not used to them. They have thinner soles making them much more sensitive than trad shoes (traditional shoes) and will need resoling sooner than other rock shoes. Long routes will often require a stiffer shoe. Boulderers often opt for performance climbing shoes.