Leadership with the mount as cohort is probably best.
I'm going to buck the trend a little bit and say Cavalier 4, just long enough to qualify for horse master, isn't too bad of a move for the right fighter. The Gendarme Archetype is pretty decent for this dip, if you don't want to have to worry about teamwork feats and tactician.
I was looking at the vitriol that is the Cavalier hate thread in general discussion and most people said that fighters, particularly the dragoon do it better.
But how to get the mount? Leadership? Class dip and boon companion?
If your GM thinks leadership is too much Ask if he allow eldrith heritage (sylvan).
One, you can't take the eldritch heritage for wildblooded stuff. Those are specifically sorcerer archetypes, so it doesn't work.
Two, I second the Gendarme dip and horse master. You want really have to sacrifice much to get what you want. You'll still be able to hit hard and get some neat abilities from the Gendarme.
Unless you need the mount to be badass combatant in its own right, you could just buy a horse.
This is my go-to answer. Also, Combat training is something that can be done in spare time, so getting a Riding Horse, then combat training it isn't that far off from taking a Cavalier Dip (though Personally, I'd dip druid, since I don't particularly like the Cavalier, and there are some decent boons to being a first level druid, not the least of which being that CLW on the Spell list means they can use Wands of CLW, making them useful as healers if needed)
Invest heavily in handle animal, stealth, survival and knowledge local or diplomacy. Consistently ask the DM if there are any local rumors about griffons or pegasi or dinosaurs or really anything else you think would make a cool mount in the area.
Track them down and steal their babies (or an adult one if you think your handle animal is up to the task) and then make with the training.
For this to work you will need a specific type of GM who runs a specific type of game.
Alternately keep your eyes open and there a good chance that at some point the party will fight something trainable and mountable. Have an agreement with the party to try and take such things down without killing them so you can train them.
Still requires a DM/game that lets you have the training time.
That's all I got.
As far as in game mechanics that let you *poof* have a strong mount all the ones I'm aware of have already been mentioned.
Well, if we are going with a bit of the traditional approach: have a druid. It has a spell that allows it to call an animal, and have its attitude start out indifferent. Eventually have someone (maybe with wild empathy) raise that to friendly, and then use handle animal to train it. That will end up with a decent mount, as long as you try to keep up with the CR power creep by replacing your mounts every so often in this manner.