Riding and Vehicle Skills

Once during your move, you can mount or dismount a creature that is within 5 feet of you. Doing so costs an amount of movement equal to half your speed, and you must succeed a Dexterity (Riding) check equal to the creature's CR + the height of a creature (1 = Tiny, 2 = Small etc) . For example, if your speed is 30 feet, you must spend 15 feet of movement to mount a horse, therefore, you can't mount it if you don't have 15 feet of movement left or if your speed is 0. If an effect moves your mount against its will while you're on it, you must succeed on a Dexterity (Riding) check equal to the effect that caused it to move or fall off the mount, landing prone in a space within 5 feet of it. If you're knocked prone while mounted, you must make the same saving throw. If your mount is knocked prone, you can use your reaction to attempt a Dexterity saving throw with an equal CR + Size to dismount it as it falls and land on your feet. Otherwise or on a failure, you are dismounted and fall prone in a space within 5 feet it.

While you're mounted, you have two options. Naturally, all mounts act independently. You can attempt to control the mount by attempting a Dexterity (Riding) check equal to the creature's CR + 5. This DC is halved if it has been trained to accept a rider. Domesticated horses, donkeys, and similar creatures are assumed to have such training. The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.

An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes. It might flee from combat, rush to attack and devour a badly injured foe, or otherwise act against your wishes. In either case, if the mount provokes an opportunity attack while you're on it, the attacker can target you or the mount.

An animal pulling a carriage, cart, chariot, sled, or wagon can move weight up to five times its base carrying Capacity, including the weight of the vehicle. If multiple animals pull the same vehicle, they can add their carrying Capacity together.
The DM of your campaign should be able to decide which creatures are deemed rideable and how, within reason.

Mounts other than those listed here are available in fantasy gaming worlds, but they are rare and not normally available for purchase. These include flying Mounts (Pegasus, griffons, hippogriffs, and similar animals) and even aquatic Mounts (giant sea horses, for example). Acquiring such a mount often means securing an egg and raising the creature yourself, making a bargain with a powerful entity, or negotiating with the mount itself.

Barding: Barding is armor designed to protect an animal’s head, neck, chest, and body. Any type of armor shown on the Armor table can be purchased as Barding. The cost is four times the equivalent armor made for Humanoids, and it weighs twice as much.

Saddles: A Military saddle braces the rider, helping you keep your seat on an active mount in battle. It gives you advantage on any check you make to remain mounted. An exotic saddle is required for riding any aquatic or flying mount.

Fighters, barbarians, rangers, druids, and gunslingers can gain proficiency in the Dexterity (Riding) skill at character creation or upon multiclassing instead of gaining a different skill.

Vehicles have a number of positions detailed in their equipment page. A pilot or driver is able to drive the vehicle as an action, increasing or decreasing its movement by up to the vehicles movement speed. For every full acceleration total added to the vehicles movement speed, it must drive "forward" 5 ft. for every 5 ft. it wants to move to the "right" or "left", increasing by 5 each distance on its turn (i.e. a vehicle that has accelerated 5 times must move 25 ft. forward to move 5 ft. to the left, but can move 10 ft. to the left on its next 25 ft.). If the vehicle's turning speeds become equal, it may choose to change its "forward" direction to the direction it was turning. For every 5 ft. above this limit, all creatures in the vehicle must succeed a Dexterity saving throw or Intelligence (Vehicles) or Intelligence (Ships) skill check or become prone, as well as be ejected from the vehicle.

In the case of vehicles with sails or balloons, the pilot only determines acceleration, with the sail operator determining how much they turn. Wind and water movements may also affect direction. A light current might only pull a ship 5 ft. forward regardless of turning, while a supernatural whirlpool might pull it 100 ft. forward and 50 ft. to the left.
If vehicle crashes into a large object, it comes to sudden stop and takes 1d6 bludgeoning damage for each 10 ft. it moved since last turn (max 20d6). Creatures the vehicle must attempt a DC 15 Strength saving throw or Intelligence (Vehicles) or Intelligence (Ships) skill check, taking an equal amount of damage on a failure or half as much on a success.
When vehicle enters a creature’s space, they take 1d6 + the vehicle's Strength modifier bludgeoning damage for each 10 ft. vehicle moved since last turn (max 20d6). As a reaction, they may attempt a DC 10 + 1 for every 10 ft. above 20d6 damage Dexterity saving throw, taking no damage on a success. If the creature is not at least 2 sizes smaller than the vehicle, it crashes as per above, decelerating a number of feet equal to 5 x the creature's size category above the limit ft.

Vehicles have ability scores as follows:
Strength represents the vehicle's size and weight. Dexterity represents the vehicle's ease of handling. Constitution represents the vehicle's durability and construction quality. Wisdom, Intelligence, and Charisma represent the same concepts as they would for creatures, and thus should typically be set to 0. Unless otherwise stated, vehicles are immune poison and psychic damage, necrotic damage if they are crafted from non-organic materials such as metal or stone, and the blinded, charmed, deafened, exhaustion, frightened, incapacitated, paralyzed, petrified, poisoned, prone, stunned, unconscious, berserk, bleeding, bloodlust, cleanse, confused, death-like state, inebriated, rasped, spasmic, pained, and taunted conditions.
Fighters, bards, rogues, and artificers can gain proficiency in the Intelligence (Vehicles) or Intelligence (Ships) skill at character creation or upon multiclassing instead of gaining a different skill.