Why are the concepts astral projection and astral traveling considered synonymous? Aren't they two different things? What's the difference anyway?
Projection is the exit of the mind or soul from the physical body, Travelling is using that psychic body to move around to visit far away places. They can be used interchangeably for most people, because I don't really know anyone who can Project and then does not Travel.
Furthermore, what is one sure-fire method to be able to get my soul-self out of my physical body? Can psychedelics help? Or, what psychic exercise should I do to make sure I get out of this physical body and get to travel anywhere and elsewhere?
Read the book Astral Dynamics. There are old school, Traditional Witchcraft herbs, salves and so on that can help with projection, but they aren't really necessary and very few people have the training and wisdom to use those sacred things correctly. You also don't really need them if you go through the correct training process, which you can receive in any number of ways. If you just want to practice this one skill, read Astral Dynamics. You can also practice Initiation Into Hermetics and slowly train your whole being until you learn Mental Wandering and Astral Projection in the later part of the system, and several other systems of magic involve lessons on this art. You can also practice Lucid Dreaming, which I consider to be fundamentally different from psychic projection despite the claims of many others. Still, it certainly can be used as practice and as a platform for external projection.
Just pick the one you want to start with and start practicing.