A QUICK RIDING SPEED TUTORIAL

a) You can level 1 slot per day at a stable from the STABLEMASTER for a small amount of gold
60 Speed
60 Stamina
60 Capacity

b) Buy RIDING LESSONS from the CROWN STORE with CROWNS (This will cost you ACTUAL MONEY to buy the crowns)

c) Have RIDING LESSONS GIFTED to you from another player.
You give that player in game gold and they use their crowns to send you the gift. The PC EU exchange rate is app 1 crown per 1900 gold

d) Buy riding lessons with SEALS OF ENDEAVOR currency
from the Seals of Endeavor section in the CROWN STORE at 360 Seals per riding lesson

e) Buy riding lessons from the CROWN STORE with GEMS.
The Riding Lessons are listed under the 'Fine Rewards' category in the crates, and items in that category are 5 gems a piece. (ty Grizbeorn)

f) Boost your RIDING SPEED with CP, SETS, PASSIVES, MUNDUS STONE and ABILITIES
CP GREEN TREE - Gifted Rider & War Horse
SETS - Adept Rider (Craftable)
PASSIVES - Alliance War Tree - Assault - Continuous Attack (As a raw level 10 player, you can do the full tutorial in Cyrodiil, including the siege, to you get rank 3 no need for any PvP .(ty Brrrofski)
MUNDUS STONE - The Steed
ABILITIES - Alliance Tree - Assault - Rapid Maneuver

g) You can HITCH A RIDE on another players MULTI MOUNT that will taxi you at their level, be careful as your FALL DAMAGE still applies when the taxi mount jumps off stuff.

h) Unlock the CONTINUOUS ASSAULT passive in the ALLIANCE WAR - ASSAULT SKILL TREE
Increases your Weapon and Spell Damage by 10% and Magicka and Stamina Recovery by 20% for 10 minutes after you capture a Lumber Mill, Farm, Mine, or Keep. Gain Major Gallop at all times, increasing your Mount Speed by 30%.

Doing a battleground, even if you lose, and the intro quest to Cyrodiil will get you to rank 3 for continuous assault, takes about 10 minutes and make such a difference when riding in Cyrodiil.