How do I get a mount as a paladin?

Solution 1:

You will need to obtain the Summon spell from your Paladin Trainer at Level 20.

At Level 40 you can get your next mount in the same way at your Paladin Trainer.

Solution 2:

When a character reaches level 20, they'll be able to get the Apprentice Riding skill from any Riding trainer (ask a guard where to find one in a capital city, like Stormwind). Then, you can purchase any mount and use it by going to your mount character pane and selecting it.

However, since you're a Paladin, check with your class trainer at level 20 to get the Summon Warhorse ability (if you're not Draenei or Tauren), the Summon Exarch's Elekk ability (if you are a Draenei), or the Summon Sunwalker Kodo ability (if you are a Tauren). This will grant you the Apprentice Riding skill mentioned above and a special, class-only mount. You can then purchase regular mounts if you so desire.

More info:

• One if by Zeppelin: Two if by Boat -- Traveling Azeroth
• Wowpedia article on "Mount"