Any character can try to use "animal handling" skill, but if that allows you to tame a beast it depends on the situation (and the DM). Usually is very difficult and train an animal takes a lot of time, you cannot magically make wolf you met your friend, but maybe you can tame a horse that is a domestic animal. They also can refuse to follow your orders if they´re against their nature or if they do not feel like doing it (that happens a lot with felines. /joke) There are no clear rules for it in the manual, to be honest.(As usual, they give "guidelines")
Spells like "speak with plants/animals/etc allows you to communicate with intelligent creatures and you can try to convince or coerce them to help you
There are spells that allow you to charm/dominate beasts and monsters, but they have a limited duration.
There are a few exceptions, like the druid spell "Awaken": when the spell expires, the creature can choose to join you, but only if it wants to. It has to be willingly.
There´s also summoning spells for everything, from find familiar to summon beasts, feys, undead, demons, elementals, etc. In 5e they´re very useful, since they last for an hour (but they´re concentration spells).
