Minecraft: Xbox 360 Edition

Go into the Tutorial world. Turn left just prior to the Beacon, and head into the farm area, up a step or 2 and enter the horse's paddock/area. Walk up to the nearest horse and hit left trigger to sit on its back. You'll get kicked off but keep getting back on the same horse until you see hearts appear over its head. Once that happens, you'll pop the achievement.

Thanks to Dynamite Andy for the extra info.

The above solution works except there's an extra step as of the newest update: you have to feed to horse wheat first. Once you've fed it one (with LT), you can then follow the steps in the other solution (keep pressing LT, it will keep kicking you off, persist until hearts appear, achievement pops).

This is correct as of May 2020 (just unlocked it) and I'm pretty sure Mojang aren't giving the 360 version of this game any more updates so you should be good from now on!

Quick update as of July 2023 to the above guides on current location. In the current Tutorial world, Grab wheat at the beginning (walk straight ahead from tutorial start until the diamond pickaxe is above, turn left and up the stairs where there is a storage chest with wheat). Then go back towards the diamond pickaxe, straight across to the stairs up. There are horses there.

As the other guides have noted, you'll just need to feed the horse or llama wheat (LT with wheat selected from your inventory), then change inventory to an empty hand, and LT to get on. Likely it will kick you off one-three times, then eventually you should see it show hearts and the achievement should then unlock after you see the hearts. Hope that helps!

In case you’re exploring, I also found a Llama at [X: -96, Y: 80, Z: -395].