Each week, you can train your horses up to 5 times in a discipline. The training boost (shortened TB) will allow your horse to gain more stats with each training. For example, a foundation with 0 TB might gain 20-30 stats per week while a horse with much much higher TB might get 100-200 stats per week. This means that horses with high TB will reach higher Performance Stats (PS) in their life time than horses with a low TB.

In order to get higher TB, you need to level up your horses before breeding them. The formula is the following: (Mare Level + Stallion Level) / 10 = Foals TB. To clarify, for example if your mare is level 36 and your stallion is level 36 too, the foal will have a TB of 7 (36+36 = 72/10 = 7.2 then rounded down to 7). If your mare is level 33 and your stallion is level 36, the foal will have a TB of 6 (33+36 = 69/10 = 6.9 then rounded down to 6).

Keep in mind that horses with higher PS will have access to higher discipline show 'ranks' which reward more experience, meaning your horses reach higher levels, thus foals will have potentially a higher TB.

Don't forget to train your foals before they turn 2 to unlock their full TB.

Ok, thanks! Where do I train them, and what is the best way?

You can go to 'Town' > 'Community Training', you'll find training centers.  Try to look into centers with all the training areas (indoor arena, outdoor arena, outdoor course, racetrack). Depending what discipline you train your horses in, you'll need a specific area. There is no "best way" to train your horse, just try to train them as soon as they can be trained to build up your horses' PS.

Note that halter shows don't depend on PS so you can still go through the halter show way to level up your (purebred) horses and have foals with better TB. You can try your horses in halter shows, see how they far there and if they do well, keep entering them in halter shows.