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:
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.
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.
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.
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.
