What is the procedure in order for me to reserve all copyrights of my songs? Is there an authority I need approval from? Is it something that I can do myself?
Do nothing. Your work is protected automatically, as ensured by (most notably) the Berne Convention, which is ratified almost everywhere in the world. Registering your work at a copyright office, mailing it to yourself in a sealed envelope, and marking it with 'All rights reserved' etc. are not requirements.
Only if you want to relinquish some of your exclusive rights, do you need to take action.
Something simple and inexpensive is to put a recording and the music into a registered / recorded delivery envelope, and post it to yourself. Then you will always have proof of when that song was born. Anyone plagiarising it at a later date can be shown proof that the song existed long before they 'wrote' it. Cheap and cheerful, as opposed to obtaining official copyright, I think.