Foal training does not improve the level bud does use energy. Levels for all horses are only improved by showing, Halter shows, carers, and exploration. Training horses of any age uses energy but does not improve levels.

To expand on Rani Harvord's point above, note that even though you won't gain levels through training, you should still absolutely focus on it. 

Levels are used in the game specifically for breeding purposes to increase a foal's s TB (training boost) multiplier. The TB is determined by the parents levels combined and divided by 10, rounded to the nearest ten. For example, a level 51 sire and level 49 dam combined are level 100, and thus produce a 10 TB foal. 

The TB comes into play after you have completed foal training or the horse has aged to 2. It does not influence foal training at all, but will heavily influence how effective regular training is for your horse.

Regular training, whether you choose a discipline or a career from the training drop down, is the only way to increase your horse's PS (performance stats) (speed, endurance, etc.). It will not increase skills or levels. A horse will gain substantially more PS from training if it has a TB multiplier than if it does not. PS are used heavily in disciplines and to a small extent in career payout. 

Foal training should absolutely still be done, but unlike regular training, it won't increase PS. The purpose of foal training is:

• Unlock the foal's TB; this cannot be retroactively done after aging up. This is the most important reason.
• Reveal the foal's personality, which is required if you want to use it in careers. This can be discovered at the vet after aging up but can be a bit pricey.
• Give the foal early access to regular training; and

• Open access to foal halter shows to give the foal a head start. This requires the foal to be purebred and registered in the foal registry, or an official game recognized crossbreed and registered in the game owned club. I think campaign foals might have some shows, too, but I just woke up and haven't participated in campaigns myself. 

   

  Random crossbreed foals just live their little lives at the stables  after foal training until they age up and are ready for one of the paths (career, halter, or discipline) or exploring. Exploring is terrible for levels, so I only really explore horses I don't plan to breed or only breed for colour (In my case, grade project horses are my only foals bred off explorers).