Having the right climbing shoes can make a big difference in how well you climb and how comfortable the climb is for you. It isn't as easy as just buying a pair of climbing shoes in your size, there are other factors that need to be taken into consideration as well to get good results.
Type of Climbing
The first thing you'll need to do is think about what type of rock climbing you'll be doing. If you're just going to be doing a few climbs in a gym that's very different than crack climbing or sport climbing. For example, those who do a lot of sport climbing are more likely to want shoes that have a downturned toe, while those who do crack climbing need to be able to fit their shoes into small cracks and thus want a flatter shoe. The downturned toe can make it easier when climbing overhangs and very steep routes because it makes it easier to grab onto footholds.
People just getting started may want to opt for an all-around shoe, perhaps one of the less expensive ones, as it may take some trial and error and people who aren't very good at climbing end up ruining their shoes faster. Once you learn the proper technique and get a better idea of the type of climbing you prefer, you can get a better quality shoe that's geared toward that type of climbing.
Stretch and Type of Material
Another important thing to know is that most climbing shoes will stretch, and some will stretch quite a bit. This makes a big difference when choosing the proper size of shoe. For example, unlined leather shoes will stretch the most, making it so you'll want to choose a smaller size than if you're opting for lined leather shoes (which will stretch a little) or synthetic shoes (which don't stretch much at all). The type of climbing also affects how tight you'll want your shoes to be. People can perform better with tighter shoes as the grade and difficulty of the climb go up. You don't want them so tight it's too painful to climb, however. It's better to have slightly looser shoes when making climbs with varied pitches or during crack climbing, but you may want to have a stiffer sole. Keep in mind as well that sizes vary by brand, so it's necessary to try shoes on and test the fit before purchasing them.
Type of Closure
You can get climbing shoes that slip on, those that close with velcro straps and those that lace up like sneakers. Although laces give you the most ability to customize the fit of the shoe, they get worn out and you'll need to make sure to replace them often. They also take longer to get on and off than slippers. Slippers can be very comfortable, but they tend to be better for gym climbing than for heavy-duty outdoor climbing. Velcro shoes fall in between these two types, as they're faster to get on and off and relatively comfortable, but perform better and fit snugger than slipper shoes.