Are climbing shoes supposed to hurt at first?
Climbing shoes are not meant to hurt at first, but it's very likely that they will. New climbing shoes can have pressure points that will go away after the break in period.
A strong misconception in the climbing community is that new climbing shoes must be painful, if they are painfully tight, it means you have chosen the right size. This is not true, and it's a pretty dangerous way of approaching sizing climbing shoes.
Climbing shoes should be tight, but not so tight that they cause throbbing pain, or that they cut your blood circulation. They should just be very snug. Your feet should not be able to move freely within them, and you should have little to no dead space in them. This does not mean that you have to choose shoes that you can barely even get on your feet.
But, again, new climbing shoes CAN be painful in the beginning, but not to the point where the pain is throbbing or that you still feel the pain after you take your shoes off. The pain should be minimal.
It is very common for them to hurt, but, they are not "meant" to. They are meant to be very tight with no dead space, and this causes them to be painful. They don't always have to be painful though, it depends what you are aiming for with the shoes, what kind of climbing, what kind of routes, etc.