If you want to ride a mount in Rune Slayer, you're going to need something comfy to sit on first. Fortunately, there is such an item in the game, and it's called a saddle (who would have thought, right?). Best of all, it works on all mounts (except aquatic ones), and you don't even have to equip it. If we've peaked your interest, here's exactly how to get a saddle in Rune Slayer.
How to get Mounts in Rune Slayer

First things first, you need to kill enemies and do quests until you get to level 20. This doesn't take that long, and you can get there pretty easily while you're in the Wayshire (the first town in the game). You can get the quest from Jimmy, the Stable Master, who is sitting in the main square. You may also want to consider taming your pet prior to doing this quest, but that's not necessary.
After speaking with Jimmy, accept the quest "Jimmy's Delivery." To finish this quest, you simply need to walk from Wayshire all the way to Ashenshire and talk to the stable master there. The trip shouldn't be too dangerous for a level 20 player. Simply exit the Wayshire gate leading out to the Greatwood Forest and go straight (turn a bit to the right when you hit a mountain). Ashenshire is on top of a huge tree, which you can reach by climbing up a rope.
The Stable Master at Ashenshire (Madonna) is the first NPC you come across as soon as you climb up the rope. Hand over the package to her, go back down the rope, and return to Jimmy in Wayshire. Speak with Jimmy, and you'll complete the quest and receive the saddle. It won't show in your inventory as an item; it's just something you'll have forever and ever.
How to ride mounts in Rune Slayer
Once you complete the "Jimmy's Delivery" quest, simply call out your mount (or tame one if you haven't already), walk up to it, and you'll get a button prompt to mount it. That's it, enjoy riding like the wind.
Remember, unless you are playing as a Beast Tamer, you won't be able to tame every pet in the game. With the re-release, the number of pets all classes can tame has been increased, but the best pets are still only available by Beast Tamers. For a complete list of tamable pets, check out how to get and tame all Pets in Rune Slayer.
That's all there is to it. Have fun playing, and if you want to stay up to date with Rune Slayer, check out the Official Trello and Discord.