You have a few options:
- The Animal Handling skill (PH p.178) usually applies to domestic animals, but would still be relevant to the lengthy and difficult process of raising a wild one.
- The Ranger's Companion class feature (PH p.93) is limited to Medium creatures, but if your character is a Small race, you could ride it.
- The level 1 spell Animal Friendship (PH p.212) will tame a beast with Intelligence less than 4 for 24 hours if it fails a Wisdom save.
- Taking the Mounted Combatant feat (PH p.168) gives you a strong metagaming argument for convincing your DM to let you have an interesting mount.