Do you have to wear bouldering shoes?
While you technically can climb in regular sport shoes, and even with no shoes, it is highly recommended to climb in climbing shoes to not only help you climb better, but also to prevent injuries. In some climbing gyms, it is a requirement to climb in climbing shoes.
Climbing holds are very rough and can damage your normal sport shoes if you climb in them often. Climbing shoes on the other hand are made specifically for these rough surfaces. They are made to handle gritty rocks and climbing holds and to not get ruined by them.
Climbing shoes will also help you climb better and give you a much better experience overall, as long as you choose the correct size for your climbing shoes.
Most climbing gyms have rental climbing shoes that you can use so that you can get used to the type of shoe before spending a lot of money on new ones.