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You have a few options:

1. The Ranger's Companion class feature (PH p.93) is the canonical way to do this and more with a wild animal, but requires 3 levels of Ranger. It's limited to Medium creatures, but if your character is a Small race, you could ride your companion.
2. 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.
3. 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.
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 #2...

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.

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.

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 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.
• 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.
• 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...

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.

You have a few options:

1. The Ranger's Companion class feature (PH p.93) is the canonical way to do this and more with a wild animal, but requires 3 levels of Ranger. It's limited to Medium creatures, but if your character is a Small race, you could ride your companion.
2. 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.
3. 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.
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 #2...

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.

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 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.
• 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.
• 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.

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 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.
• 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.
• 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...

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.