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If you click on the breed page for your horse you can see which skills are the best for the breed and train those.

for this obj it has to be the horses skill that is highest at the moment you are trying to do this obj. it's not always going to be the top skill for the horse overall. for example the horse below (just a random horse in the sales) the top skill for based on genetic potential is speed but the top skill right now is dressage. if you were going to use this horse for this obj you need to do dressage.
https://www.howrse.com/elevage/fiche/?id=90209932
to add: my info is correct this obj is very specific. it MUST be the skill that is highest at the moment you are doing the obj. this is often the highest one based on GP but not always.

I think training in his 'best skill' is not necessarily the one that is highest at this very moment. In my opinion, what they mean here is the skill in which this horse is best once he is fully trained.
You can find this by clicking the Genetics tab.
For the same horse mentioned in the other answer (https://www.howrse.com/elevage/fiche/?id=90209932) this would be speed.

But to make it easy, you can choose one of your horses that is not fully trained and do 1 hour training in each of it's skills.
Then the objective should be completed, whatever the best skill may mean ;)
The objective does not say 'fully train', but just 'train'. That means 1 hour of training in that skill should do the trick.
So for te mentioned horse I would train speed and dressage.