Riding lessons increase the riding skill of your character, not the performance of your mount. As such it applies to all mounts you might use with your current character, but if you make another character, that character starts from square one again.

Go to Cyrodiil and do the intro quest for a permanent +30% riding speed boost from an Assault skill line pssive. No actual PvP needed.

There is also +10% riding speed to be had from a green CP passive, if you have a lot of CP (it's fairly high up).

Don't forget you can train daily. It's tempting to go for some riding speed on a new character; I might certainly spend some Endeavor points on that, but both Crown and Endeavor lessons are terribly expensive for what they are really.

The 30% boost from the Assault passive Continuous Attack is available once you reach level 10. You just have to do the training quest, no actual combat.

If you got your single mount as a level-up reward, you’re beyond that. Without it, you can probably run as fast as you ride, so grab it even if you never intend to PvP. You get enough skill points from the training visit for that and Vigor, if you need that as well. Both become available with the levels you get from the training itself, which typically takes you level 3.