Determine the type of climbing shoe you need. Traditional shoes are for long climbs, including cracks, face, chimneys, and slabs. They generally have a board last, fit a little more loosely than other climbing shoes, and are comfortable for many hours of climbing.
Sport shoes are for mostly face climbing, limited crack climbing, and some climbing gym activity.
They are generally slip-lasted and fit more tightly. Performance shoes are generally for overhanging sport routes, bouldering, and competition - not for beginners. Find a climbing shop with a large selection that's deep in sizes. You're going to need to try rock shoes on in person. Note that a good shop will have an artificial wall in the shop for trying shoes out.