While generally you can go climbing without climbing shoes, it is highly recommended to wear climbing shoes. Some indoor gyms also require the use of climbing shoes and forbid climbing in anything other than them.

You don't need to wear socks with rock climbing shoes, most people don't, but many people do for different reasons.

If you have blisters, wounds, or want to prevent blisters, a thin pair of socks can help quite a bit with that.
If your shoes are a little bit too loose, socks could help tighten them up.
Some people also just like climbing with socks.

Climbing shoes are generally designed to be very snug around your feet, with no socks in between them and your feet, but in the end it's up to you if to wear socks or not.