I just want to ask since I'm basically new to the game: taming an animal seems interesting, so how does it work and what do I need to do it? What rules or guidelines exist around how one can tame a creature?
The fifth edition of D&D has come back to the concept of small parties, opposed to being able to have an army of minions, pets, familiars and undead creatures.
For this reason, there is no rule for turning beasts into allies, as far as I know, at the moment. You can calm them down with Handling Animals and you can befriend them with spells like Speak with Animals, but there is no way to turn it into a companion.
There are some class mechanics that allow you to have a companion. The closest to what (I think) you asked is Ranger's animal companion in the Beast Master path at 3rd lvl. Wizards also have the spell Find Familiar, which would be similar.
Even for mounts, as far as I know (and I've quickly checked the PHB and DMG) there isn't an official way to tame them, you just buy them already tamed or receive it as a reward/gift/something.
Having no official rules for it doesn't stop you (as a DM) of making them up, or you (as a player) of talking to your DM and check what you can do. Maybe you just want a pet wolf that walks along with you for flavor, but is completely irrelevant combat-wise (he is too scared to fight, maybe?) or a pet mouse (that is actually a starting item for some backgrounds). Maybe even a fighting companion won't be that bad if the character puts enough effort and the DM thinks it may work.
TL;DR: There are no official means. If you are a player and wants to have an animal friend, talk to your DM and say exactly what you said here (you found it interesting and want to know if and how you can do it)