Levelling up

I'm levelling up really slowly and I'm trying to get to level 6 so I can breed. Any advice on the best and fastest ways to level up are much appreciated :)

The only way to level up is with horses, so the more you have the faster it will go.
Make sure you use the training arenas every day, and either show or use the riding school as these all increase your experience.

Some horses will do better at showing while others may do better with the riding school, if you want to get the max amount of exp then you'll want to pay attention and put horses who show poorly in the schools. Also by using riding schools with the same specialty of the horse will give you a small boost than with other specialties.
I think shows with a higher amount of entries give out more stats as well, though that may only apply for the show winnings. I haven't really paid attention xP

Training your horses gives member experience for using the correct level and correct discipline arena, so make sure you use the right ones!

Riding school is the most consistent way to gain experience. if you use Group lessons you can get 30 EXP per hour, so 300 per horse if you do all 10 hours, 330 if you use a school with the right discipline.
Showing is less consistent but can really pay off, you can get 30-125 exp per show for getting first place. You still get experience when you don't place but it's much less. Experience given is based on the amount of entries in the show and where your horse places, so a better horse will get more exp!
And only shows with 5 or more entries have a chance at giving out stats, but all shows give EXP, show points and money.
Let us know if you have more questions!