Hey there. I am a pretty new player too, I started about 1 month ago.
There are a few way to train your horses. 
It can be a little complicated. 
If your horses have good skills (The progress bars on the stats page) and good breed standard (The number and colors at the bottom) you should show them in halter shows. To do this you have to have your horse registered with the breed registries.  

If your horse has good breed standards and good PS (Preformance stats) then show them in the discipline that matches their stats.

You can also train your horses up by riding them around the hoofbeat plains.

I highly recommend selecting the help tab and reading the documents on the ‘wiki’ site.
Once you get the hang of the game it's really fun.

Feel free to look through my barns. I have my horses organized by what I use them for and a couple of my horses have their breed variance in their notes at the bottom of their profiles. 
I keep quarter horses but I have been collecting a couple other breeds to join a cross breed campaign.

that's leveling, it increases the level of the horse by gaining experience, while training is what increases performance stats. to train a horse it needs to have a discipline and/or career set. then click on the dumbbell icon on their page underneath their picture and select a training center. alternatively, you can go to Town → Community Training and select a training center from there.