First off, you should know what to roll. Four races -- human, dwarf, draenei and blood elf -- can be pallies. If you're Horde, your choice is limited to one. Alliance has more choices, so you need to look at the racial of the classes and what races you've already played. If you haven't made a draenei yet, the 1-20 content will be new to you. Plus, the 1% hit aura is pretty nice. The human Diplomacy racial is killer if you're planning on doing heroics, and both Sword and Mace specialization come in handy if you're going Ret. As for dwarves ... well, dwarves have Stone form, which is nice for PvP. There aren't a whole lot of benefits to being a dwarf paladin, but they are kind of cool-looking, especially female dwarves.
The skills you begin with are very limited. You start out with the armor-adding Devotion Aura, the holy damage Seal of Righteousness, and the healing spell Holy Light. This combo gives you fairly high survivability, but little damage and no crowd control. As such, it's best to take on one enemy at a time until you're comfortable with the class. Just put on Seal of Righteousness and Devotion Aura and hit stuff, and heal yourself when you get low on health. But a couple stacks of food and water to help you along. Sell all of the vendor trash you get to make the cash for your skills.
Depending on what you think, you'll start in a monastery (human), an icy plain (dwarf), a crashed spaceship (draenei) or a beautiful city (blood elf.) The Alliance races begin by using maces, while the blood elves start out with swords. You'll be unable to use some of the other weapons until you make the route of big cities, so it would be best to stick with your given weapon type at the beginning, or make your way to another capital if you're really desperate. You'll get a couple quests to run around, find your paladin trainer, and talk to various people, and then you're going to start your questing.
From here on, we'll go level by level to the new things you get and how they should affect your play style.
Level 4: You receive your first blessing, Blessing of Might, and the Judgement ability. Now, you'll be able to Judge your Righteousness for extra damage. Don't forget to reapply it! A macro can help wonders there. Also, keep BoM up constantly, as it'll help out your meager damage.
Level 5: Time to pick your professions. If this is your only character on your server, it's probably best to pick up two gathering professions, like mining and herbalism. If you're going ret in end-game, mining and blacksmithing will get you some awesome weapons. Tanks and healers can pick some non-weapon and armor professions, like jewelcrafting, enchanting, engineering or alchemy.
Level 6: You get your first bubble, Divine Protection, and your second seal, Seal of the Crusader. Your fights should now begin with sealing and judging Crusader before turning on Righteousness, since Crusader increases your holy damage. At this point in time, Divine Protection should probably be used to get a heal off (easy with your new rank of Holy Light) or run for it.
Level 8: Not too exciting of a level. You get Hammer of Justice, which stuns your opponent for 3 seconds, and Purify, which removes a disease and poison effect.
Level 10: Now we're jamming! Your Seal of Righteousness and Devotion Aura go up a rank, and you get two more abilities: Lay on Hands, which heals your target to your maximum health for the minor cost of all your manna and a one-hour cool down, and Blessing of Protection, which can serve as a second bubble at this level. Later on, it's more useful to bubble party members. Both of these are good escape mechanisms if you find yourself in an "oh, crap" moment.
Level 12: You've probably moved from your starter area to the next (10-20) place, but don't forget to go promptly back to your trainer when you level up. At 12 you learn new ranks of Blessing of Might and Seal of the Crusader. You also get your first class quest, to learn the razz spell Redemption. Hordies will be summoned to Silvermoon City and asked to light a brazier in a Ghostlands cave, where you're attacked by another Blood Knight. Then you need to fill up a vial with the power of the Naaru M'uru and ressurect the guy who attacked you (who's been moved to an inn.) It's pretty easy and provides a good overview of Blood Knight Lore. Draenei will be called to the Exodar to read the Tome of Divinity, and then use a one-time rez spell to rez a furbolg shaman.
Humans and dwarves have it a bit tougher. They'll be sent from their paladin trainer to Duthorian Rall in Stormwind or Tiza Battleforge in Ironforge. Both these questgivers ask you to go help someone in need. This means John (in Ironforge) and Stephanie (in Stormwind) Turner, who collect linen cloth for orphans. Orphans in linen sweatshops? They never tell. Bring them 10 linen cloth, and you'll be shuttled around for a while before being asked to resurrect a dead guy in Dun Morogh/Elwynn Forest, and retrieve a letter from a Defias member or Dark Iron Dwarf. Then, and only then, will you get your Redemption.
Level 14: You get a new rank of Holy Light and two new skills: Blessing of Wisdom, which increases your mana regen, and Righteous Defense, which is basically the paladin taunt. Use your judgement on BoM vs. BoW. If you don't have any Judgement, why are you playing a pally? *rimshot*
Level 16: At 16 you learn Retribution Aura, which should immediately replace Devotion Aura, and Righteous Fury, another skill that is good only for tanking at this level.
Level 18: A new Seal of Righteousness and a longer Divine Protection will help push you to 20, as will Blessing of Freedom (removing all movement-impairing effects) and Spiritual Attunement (which gives you manna back when you're healed.)
Level 20: Congratulations, you're here! Enjoy your prizes: Exorcism, which hurts undead and demons; Consecration, an AOE attack that does holy damage over a few seconds; Flash of Light, a quick heal that heals a lesser amount than Holy Light; and a new rank of Devotion Aura. Depending on your talent points, you can also pick up Blessing of Kings, which increases stats by 10%, or Seal of Command, which can proc for major holy damage when you're fighting. If you took my advice and went Ret, you'll be getting the latter.
You've also got a new class quest to go on. The Horde will be asked to undertake the Second Trial and kill more Blood Knights near Silver moon for the skill Sense Undead, which is exactly what it sounds like. Allies will go instead to Westfall and protect an innocent woman from the Defias thugs for the same skill.
You'll then be asked to start what might seem like an awesome class quest for a great weapon, but will instead end up destroying your soul. Hordies need to go to Deatholme, RFC, SFK and BFD to get the Blood-Tempered Ranseur, and Allies must trek to Deadmines, BFD, SFK and an elite area in Loch Modan for Verigan's Fist. Unless you have a group of friends or a level 70 to run you through, these are so not worth it, even though the Alliance weapon will last you for some time.
After 20, the whole world will open up to you. Head to a contested area and start questing, because even with the leveling buff, it's going to be a long way to 70.