I’m not one to think the only way you should train is with your bodyweight. Sure you can do that and get great results, but in order to be the most athletic you can be, you’ll have to look toward other disciplines.
All the various training tools have advantages and disadvantages to them. With certain implements you can train in ways that cannot be duplicated elsewhere. Why not take the best of everything?
That being said, first and foremost you should know how to use your body. I don’t think you should begin to use anything else without a base level of expertise in your own bodyweight.
You’ll only ever live with your own body so learn how to use it well. There are certain advantages with bodyweight exercise, not the least of which is you don’t need any equipment and it can be done anywhere.
Just look at a gymnast. They do bodyweight exercises and no one can doubt the strength they have. You may never aspire to do a planche or an iron cross but understand just how difficult these exercises are. Controlling a barbell is one thing but to be able to manipulate your own body in time and space requires much more coordination and strength.
A large benefit of doing bodyweight exercises is the entire body will be worked in unison. To do a pushup you must maintain a certain degree of tension in the midsection so that you can do the movement up and down. If most people were to try to hold a handstand against the wall they would come tumbling down. They do not have the control necessary to keep their body together. This same effect can be achieved with weights but it is not always so readily apparent.
Since you are the resistance, it is easy to get inside the weight. A beginner will have a disconnected mind when lifting weights most of the time. With bodyweight exercises you can link up your body and mind with more ease. More experience with focusing within will allow you to carryover the effect whenever it is needed.
The wide variety of bodyweight exercises can be used to build maximal levels of strength, maximal levels of endurance and conditioning as well as flexibility, mobility and balance. By manipulating the manner in which you train you can achieve any goal you want, losing fat, gaining muscle or just getting in better shape and stronger.
There are calisthenics, jumping rope, hill sprints, muscle control, isometrics, hand balancing, gymnastics, yoga, even feats of strength using the body alone, and much more.
How you train with bodyweight exercises is the important thing. When you truly learn how you can use your own body you’ll be able to make one pushup so difficult you can’t do it or be able to crack off 250 with some ease.