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.
The amount of time it takes climbing shoes to break in and to stop being painful depends on a few things, such as the specific shoes, their size, the material of the shoes, and how often you climb in them.
The most important thing is to, first, get the right shoes for your feet, and the right size. If you get shoes that just don't fit your feet, it won't matter what size you get, they just won't be a good match for your feet.
The same goes with sizing. Even if you find the perfect shoes that match the shape of your feet perfectly, but you don't get the correct size, it won't matter that they fit like a glove.
Both things go together and both can determine how long the break-in period will last, and how long the shoes will hurt, if they even will.
Here is some more information on breaking in climbing shoes.
