The early going: Level 61-62
There's no real need to rush back to your trainer just yet, because all you'll learn at Level 61 is a new rank of Seal of the Crusader. Unless you've wrapped up all the available quests in the area, there's no urgency to leave the barren wasteland of Hellfire. There are plenty of demons here, so you'll have an extended feeling of usefulness. Make your way to Shattrath City in Terokkar Forest and set your hearthstone at one of the Inns there. It's also a good time to consider whether your Paladin will side with the Scryers or the Aldor. Do the quest chain along the road to Shattrath, as it's pretty quick and in the area. Don't forget to get all the flight paths in Hellfire, as well. Around these levels will be a good time to venture into Zangarmarsh and meet the Cenarion Expedition. EDIT (thanks to cold40): Protection Paladins will also do well to make friends with the little folk from Sporeggar. It's relatively easy to get to Honored and pick up an ugly turtle shell, an excellent tank shield.
When you hit Level 62, you should rush back to your trainer and pick up the one skill that will serve you throughout all your remaining levels no matter what spec you are -- Crusader Aura. Remember the rush you felt when you graduated from your old, busted joint to the new hotness? Crusader Aura is something like that. It's the WoW equivalent of the Turbo Boost. At 62, you'll also learn a higher rank of Blessing of Anti-Sheep, as well as a stronger Holy Light.
Be sure to complete the quest chains in Hellfire Peninsula for blue rewards, as well as venture into Hellfire Ramparts and The Blood Furnace for a chance at good drops and very fast experience. Weaken the Ramparts has a good (if ugly) Protection or Retribution shoulder reward while its immediate follow-up, Heart of Rage, rewards an excellent healing ring. It might be a good experience to try doing the Hellfire Fortifications daily quest to accumulate enough Marks of Thrallmar or Honor Hold to purchase Libram of Zeal when you get to Zangarmarsh. It's fairly easy to get, requiring 15 Marks, and will help most Paladins while grinding. At Level 61, you're also eligible to enter The Eye of the Storm, so try out the Battleground and complete the daily quest when it's available. You'll need 40 Eye of the Storm Marks of Honor to purchase PvP boots and 20 for the weapons, so it doesn't hurt to stock up on them as early as now.
The midway point: Level 63-67
At Level 63 you'll have a choice of either Zangarmarsh or Terokkar Forest to quest in -- finishing both zones will see you past Level 65-66, so it's alright to stick to one zone and finish up the quests there. Zangarmarsh has The Slave Pens, so it's not a bad idea to hang around the area just in case other players are LFG. Another rank of the non-scaling Shadow Resistance Aura is the only thing you'll learn at Level 63, so it's nothing to rush back to Azeroth for.
At Level 64, however, Horde Paladins get Seal of Blood while Alliance Paladins get, um, Seal of Vengeance. These are the spells that distinguish between the Paladins from either faction, so it's a point of pride. Horde Retribution Paladins should rejoice while the Alliance can probably admire the pretty new icon on their hotbar. You can also venture into the Underbog and complete two quests with excellent rewards. The Everlasting Underspore Frond is a wonderful leveling item that saves players a good deal of money, while the Mario Bros.-inspired Power Infused Mushroom allows Paladins to grind just a little bit longer. The dungeon also has excellent Retribution Paladin drops, such as Truth Bearer Shoulderguards, Studded Girdle of Virtue, and the Doomulus Prime twin, Hatebringer.
At Level 65, you'll learn Blessing of Wisdom and Greater Blessing of Wisdom. It's also a good time to venture into Nagrand and Blade's Edge Mountains. Both zones have excellent quests, although Nagrand does have the Nesingwary series of killing quests which can be done in quick succession. Kill quest series' are the most XP-rich quests in the game, so head on over to Nagrand and collect poop while you're at it. Completing quests in Nagrand, Blade's Edge, or even in Terokkar Forest should push you to Level 66. You'll learn new ranks of old spells, including the all-important Spiritual Attunement (Rank 2).
You should complete the Ring of Blood series of quests with a good group (or helpful Level 70 guildies) as each step in the quick series awards 17,450 XP and the final quest awards decent gear for Retribution or Holy Paladins with the Honed Voidaxe and Mogor's Anointing Club. Both weapons should last players well into the late 60s or even as starter weapons for the Level 70 instances. Blade's Edge and Nagrand are loaded with enough quests to take players up to 70 if both zones are completely quest-scoured. Terokkar Forest has Mana-Tombs and Auchenai Crypts for those who can get groups for it. Mana-Tombs' Pandemonius -- an easily farmed first boss -- drops the Shield of the Void, one of the best pre-70 shields available for Tankadins. It's not quite worth mentioning that Paladins can learn a new rank of Seal of Wisdom at Level 67.
Almost there: Level 68-70
At Level 68 and 69, Paladins learn new ranks of old spells but at least get the strongest Hammer of Wrath available. Around these levels, players should make their way to Netherstorm or Shadowmoon Valley. Demons teem in both zones, making them extra-efficient places for Paladins to farm. Your choice of Aldor or Scryer will come into play here as there are faction-specific quest chains in both areas which culminate in different quest rewards. The Scryer base in Shadowmoon Valley, or the Sanctum of the Stars, is generally considered to be in a better geographical position for quests. It's also closer to Netherwing Ledge, which is a boon for when you finally decide to grind for a Nether Drake and find yourself (quite often) on the receiving end of an angry Dragonkin's hammer. Quests in these areas, coupled with the higher level instances, should push you all the way to Level 70.
Life begins at Level 70
It's probably a good idea to ding in Shadowmoon Valley as this is where players can train to ride flying mounts as well as purchase them. Despite having Crusader Aura, traveling on foot (or hoof, as the case may be) pales in comparison to flying. Remember how I said you should save up that Gold? This is the time to spend it. If you managed to save up 6,000 Gold, skip purchasing the normal mount. You can hit 70 through finishing the many quest lines in the zone, or do something clever like Turpster and his Dingstravaganza. At Level 70, a whole new world of play opens up. Level 70 instances, raids such as Karazhan, and even Arenas. Flying opens up areas previously unreachable, opening up quests with new factions such as Ogri'la and the Netherwing. And of course, Paladins learn the highest rank of spells available as well as the Level 70-only Avenging Wrath. It's a relatively easy rush from Level 61 to 70. Enjoy being max level while you can, because the grind will probably have to happen all over again in Northrend.
Are you a fan of the Light and all that's Holy? Or are you a fan of dispensing Justice? Read up The Light and How to Swing It and learn about the many things that make up a Paladin. More importantly, go check out WoW Insider's Leveling Guide to all classes! It's a hefty read that'll get you swingin'.