There is so set way to do it. Some people level based on points. For example:

0-20 - Beginner
21-50 - Intermediate
51-100 - Advanced

OR

0-50 - Beginner
51-100 - Intermediate
101-200 - Advanced

Doing it that way depends on how fast you want to be moving your horse up/how fast you are aging your horse etc. Some people who play the game at a faster pace might use the lower point system to level so that they can retire their horse faster for breeding.

There is also leveling based on experience (my own personal method). This is based more on the number of competitions at a particular level your horse has entered. I use this method more for the RP factor (it's more realistic to level based on experience). You can do it like so:

10 shows @ Beginner - Move to Intermediate
20 shows @ Intermediate - Move to Advanced

You can decide how many shows you want to enter before leveling up too. It's all up to you, for both ways. Both have pros and cons.