Do climbing shoes hurt at first?
Usually yes, but it should really be more of an uncomfortable feeling rather than actual pain. If you are used to climbing shoes, it may be fine, if not, you get used to it pretty quickly, assuming you didnt size down too much.
There are a few scenarios in which climbing shoes may hurt:
1. If you are a new climber, and you are not used to climbing shoes
If you are a new climber, and you are not used to climbing shoes, they may hurt at first and be very uncomfortable. Climbing shoes in general are not very comfortable, and this sometimes causes pain to seem much worse than it is for beginners. After a few sessions, new climbers usually get used to the feel of climbing shoes. It's recommended to start out with flat or beginner level shoes, as the more aggressive climbing shoes can be difficult to get used to from the start and may be much more painful than flat beginner shoes.
2. If your climbing shoes are new and they haven't broken in
Climbing shoes that are new and haven't broken in is a very common reason for painful shoes. This is especially true if you downsize your climbing shoes. Once your shoes are broken in, the pain should be reduced or eliminated completely. If it is not, your shoes may just not be the right size or shape for your feet.
3. If your climbing shoes are just not right for your feet
Another common reason for feeling pain in climbing shoes is if you bought shoes that are not the right size or not the right shape for your feet. Feet come in all shapes and sizes, as do climbing shoes. Not all climbing shoes fit all shoes. Many new climbers decide to get what most others in their climbing gym say are the best shoes, or they get what they see most others wearing. Just because most wear them, doesn't mean they match your feet, and this is something many climbers don't realize.
It's important to get the shoes that match your feet and at the right size. If you have wider feet, get wider shoes. If you have narrow feet, get narrower shoes. There are shoes that are made specifically for wider and narrower feet, such as the Mad Rock Drone, they have a high volume version for wider feet climbers, and a narrower version for narrower feet climbers.
Find what works for you, not what works for others.