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A lead can be crafted out of 4 spider strings and some slimeballs using a crafting table. You can get spider string from killing spiders, and slimeballs from killing slime. Slime spawns in Swamp biomes. The lead looks like a leash. Place the lead in your toolbar and select the toolbar slot to equip it. This places the lead around the horse. You can now walk the horse. While walking a horse with a lead, click a fence post to tether the horse. This keeps it from wandering off. Fence posts can be crafted out of wood from trees. Horse armor can be crafted out of 7 leather, iron, gold, or a diamond using a crafting table. Leather can be obtained by slaughtering cows. Iron, Gold, and Diamond must be mined to obtain. Iron and gold ore needs to be smelted in a furnace to create iron and gold bars. This opens your inventory with the horse menu at the top. You can transfer from your own inventory to the horse. The armor slot resembles the top side of a horse in the upper-left corner of the horse inventory window. Drag the horse armor from your own inventory to this slot. Place a chest in your toolbar and select the slot with the chest to equip it. Right-click on the mule or donkey's back. That will put the chest on the mule or donkey. Chests can be crafted from 8 wood plank blocks. To access the chest, right-click while riding the mule/donkey. To remove the chest, right-click while riding the mule or donkey and there will be an icon of a chest in the armor slot. Remove that chest via shift-clicking or dragging the chest to your inventory. Chests cannot be placed on a horse. A saddle is needed to ride a horse once it is tamed. Saddles are one of the few items that cannot be crafted. You can find a horse saddle in chests in dungeons or nether fortresses. You can also catch a saddle while fishing. This will display your inventory with the horse menu at the top. You can transfer from your own inventory to the horse. It's the slot that resembles a horse saddle in the upper-left corner of the horse inventory. This places the saddle on the horse. With the horse saddle equipped you can mount the horse and control it using the "W" "S", "A", and "D" keys. You can also jump by pressing the spacebar. Like the saddle, name tags cannot be crafted. They can be found in dungeon chests and within nether fortress chests. To engrave a name on a name tag, have the name tag in your inventory and right-click on an anvil. Drag the name tag to the first slot on the left in the Anvil menu. Click on the box below "Repair and name" and delete "Name tag" in the box. Then type the name you want to name your horse. Drag the engraved name tag in the slot on the right back into your inventory. This costs 1 enchantment point. Place the name tag in your toolbar and highlight it in your inventory. This places the name tag on the horse and names your horse. You will see the horses name above it.
Craft a lead. Equip the lead. Place the reticle on a tamed horse and right-click on it. Click a fence post to tether the horse. Craft horse armor. Mount a tamed horse and press E. Drag the horse armor to the armor slot. Put a chest on a mule or a donkey. Obtain a saddle. Mount the horse and press E. Drag the saddle to the horse armor slot. Locate a name tag. Use an anvil to engrave a name tag. Equip the name tag. Place the reticle on a horse and right-click.