Is it normal for new climbing shoes to hurt?
It's normal and quite common for new climbing shoes to hurt a little bit. With that said, they should not cause throbbing pain that lasts after you take them off.
Depending on the type of shoes and their material, new climbing shoes usually have a break in period, which can take a few climbing sessions to get through. During this break in period, your shoes will start to change in shape and fit, molding themselves to your feet. This process can be very uncomfortable and a little bit painful, depending on the size you bought.
Keep in mind that this break in period varies depending on the type of climbing shoe, the material it is made of, and the size of the shoe. Synthetic shoes may stretch at the most half a size, while leather shoes can stretch up to 2 whole sizes.
If your shoes are synthetic, and they are causing you pain, keep in mind that they may not actually have a very noticeable break in period, which means they may just not be right for you. You may also not be used to wearing climbing shoes that are tight, which also takes time to get used to.