There is no one correct way of climbing since every climber can have their own climbing style, and strengths. Though, there are a few general concepts that can help you climb better:

1. Use your feet - Climbing is not just an arm and back sport, contrary to what many believe. It also heavily relies on other things, such as proper footwork. If you rely on just your arms and back muscles for climbing, you will soon find out that you cannot progress as quickly as you would like. Having proper footwork is key to better climbing, as most of your weight should be on your feet, and not held by your arms.

2. Keep your arms straight - This is an important part of reserving strength while climbing. If you consistently lock off (holding your body weight with bent arms), you will very quickly lose strength and stamina, and will end up just falling and not completing the route. Allowing your arms to straighten out helps transfer your body weight to your skeleton, which reduces the tension and load on your muscles, allowing you to climb longer and reserve the much required power for later moves.

3. Technique - Sure, you can climb some routes by just pulling yourself up with your arms, but at some point, the routes will require you to have increasingly better technique. Good technique is better learned from the beginning, rather than later on after you have gotten use to relying on your strength alone. With proper technique, you are able to climb longer by reserving stamina and strength.

4. Read the route - Another key element in being able to climb hard routes and to preserve energy and power is to learn how to read the climbing routes rather than just getting on them and trying to figure it out on the wall. Trying to figure out a route while on the wall and where the next holds are and how to reach them just takes a lot of precious time that will cost you a lot of energy and strength. Reading the route and understanding it before getting on it allows you to see the holds, understand how to get to them and hold them, and gives you a general map of the route for you to follow. This helps reserve energy and strength and can be the difference between reaching the top and falling.

All climbing shoes have a break in phase. During this phase, the climbing shoes will stretch and get slightly softer in some areas. The amount at which they will stretch and get softer depends on the size of the shoe, the material, and how you climb in the shoes.

If the shoes are synthetic, the break in period may be shorter than that of shoes with leather since synthetic shoes normally do not stretch that much, they usually stretch around half a size to one size. This means that if you buy a size 9 synthetic climbing shoe, it will most likely still be the same size and fit as you bought it, even after many months of climbing in it.
With synthetic shoes, though, It's important to keep in mind that they will get softer after consistent use, but, since they won't change in their fit too much, and they will be almost as tight as you bought them even after 6 months, it's important to really get the correct size and to choose shoes that fit your feet properly. Otherwise, the break in period can take much longer, and may not happen at all if the shoes just don't fit your feet.

If the shoes are leather, the break in phase is a little bit different, since leather shoes can stretch up to a few sizes larger than when you bought them.

This is from personal experience of having 4 pairs of synthetic climbing shoes, all of which are as tight as they were when I bought them and have not stretched (from what I can feel at least) even half a size, and this is after between 2-3 years of consistently climbing in them 4 times a week. They have become softer to some extent, which reduced the feeling of tightness in some areas of the foot. The break in period for me was relatively non-existent, since they didn't really stretch, and I bought them with a pretty good fit out of the box. They started getting softer, I would say, after about a month of consistent climbing,

Rock climbing shoes are worn the same way normal shoes are worn, just without socks. Their sizing, though, is a different matter.

Rock climbing shoes are very tight in comparison to regular sport shoes, this is to provide better performance while climbing. This extra tightness is a little bit weird at first, but eventually you get used to it and the discomfort goes away.

You normally wouldn't wear socks in climbing shoes to make sure that your foot does not move around inside the shoe, which could reduce its performance.

Most climbers like to take the heel off in between routes. This is because the shoes are tight, so taking the heel out just reduces that stress on the feet and helps get you ready for the next route, and allows you to climb longer with less pain in your feet.

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.

Generally, you would not wear socks with climbing shoes unless you have a reason for it. Socks can help prevent chafing by providing an extra layer between your toes and the shoes. This could help in cases where you have new climbing shoes that are very tight or that you are not used to yet, or, you are a beginner climber, and you are not used to your climbing shoes yet. In these cases, the climbing shoes may be rubbing against your toes to the point where it actually breaks the skin and peels a thin layer off. Wearing socks can prevent this.

Other than this, though, it is usually preferred to not wear socks, since an important reason for climbing shoes being so tight is so that your foot will not slip inside of it or move around. Wearing socks can cause your foot to move around in your shoe, even in very tight climbing shoes. This can happen especially in the heel area and when you have difficult heel hooks where you really have to pull on your heel. Wearing socks can cause your shoes to not stay on as tight as they should, and could result in your foot slipping, the heel of the shoe coming off, and even in injuries if your foot slips in a risky way.

So the recommended way to wear climbing shoes is without socks. If you really want to, though, you can find a thin pair of socks to wear, just until your toes get used to the friction of the climbing shoes.

Climbing shoes are bent downwards to enable you to put more weight on very small edges. By being bent downwards, you are able to apply more force on the small edge without the shoe slipping off.

Bent climbing shoes are normally better for bouldering and overhanging routes, as they allow you to have a better grip on the holds, which enables you to pull harder on your toes and keeping your body close to the wall. All of this allows you to reserve more energy in your arms.

Though they are mostly used for bouldering and overhanging routes, they are not limited to them. There are many climbers who prefer aggressive downturned climbing shoes over neutral ones for sport climbing also.

Another element of aggressive climbing shoes is a slight twist in the direction of the big toe. This allows you to put even more force and power onto the toes, allowing you to have an even better stance on those very small edges.