This post has some ideas in it, and other relevant posts linked in the first comment:

https://horsefable.com/topic/6570/

It kind of depends on the horse - if they're strong in halter then you make the most points that way and level up more quickly. But it's easy to lose bits in halter so if they aren't consistent you may want to switch it up.  When you're on an individual horse, the last tab shows you their history - placings, entry fee and winnings. So you can get an idea of what they're best in for you.  I often don't have time to enter everyone in shows, so I rely on careers - they'll earn you bits and points. They earn more points as they get stronger in careers too.  They definitely can level up with just career but it is a bit more slow, especially in the beginning.

id have to say i agree with what @snowfalls said halter shows are a good way to level up faster but if you dont have enough funds to support said horse that cant keep up and win halter it will be hard to get points there at that point id say try doing events like show jumping, barrel racing etc. i dont know much about careers as i dont do that but id say give that a try as well? 

Thank you guys so much!