You have a few options:

1. 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.
2. 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.  This is a short term solution, but it could also be a jump-start for long-term taming.
3. The Ranger's Companion class feature (PH p.93) is *the* canonical way to tame a wild animal, but requires 3 levels of Ranger.  It's limited to Medium creatures, but if your character were a Small race, you could ride your companion.
4. 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.

**A Starting point for option #1...**

Since we're in house rule territory, my suggestion would be to treat taming the wild animal as a downtime activity (PH p.187) similar to item crafting.

* A proficient artisan can create 5 gp worth of value per day of work.
* The "Mounts and Other Animals" table (PH p.157) gives market values for various species.
* Divide the market value of the mount by 5 gp to obtain how many days of successful Animal Handling rolls are required to tame it.
* The DM chooses a DC based on how appropriate the animal is for domestication.

For example, breaking a bronco and training it as a riding horse worth 75 gp would take 15 days of successful rolls at DC 10 (easy).

It's probably not realistic, but it's playable -- quick enough for a character to actually do it, and balanced to the in-game value of skilled labor.