To speed up the leveling process, Wings of Teleportation are a must. These can be purchased from an NPC near the Auction House in Greenmont. If on a budget, stock up on Teleport: Silence since they are the cheapest port to buy at only 10 
each. When finishing a group of quests, teleporting back to Silence and then porting to the location to turn quests in can save a lot of time.
While leveling up, you will receive a Box with usually helpful items in it: gear, experience blessings, buffs, etc. Don't use the exp blessings for doing long travel quest chains. Instead, save them for quests involving killing mobs.
Most classes can pull and kill multiple mobs at once, you would just need to practice the best way to do it. Once learning a good rotation everything else becomes much easier. There are also certain mobs in each area that give more exp for grinding.
See: Experience, for a list of available features that are unlocked per level.
Arcadian Forest - Level 1-12+
Teleport: Silence available at level 6.
At low levels it's easier to complete quests and kill mobs inbetween than to grind mobs. When mob levels become green, finish up any quests that have gear upgrades and move to the next quest area.
Areas to quest: Silence (lvl 3-8), Waterby (lvl 6-9).
Tear Coast - Level 14-19
Teleport: Bellshoal and Teleport: Greenmont available at level 10.
Completing quests in Tear Coast will get you to level 18, in which you can do the first dungeon Knahswahs Prison using Dungeon Finder for a lot of exp. To queue into a dungeon, use Menu > Team > Dungeon > select the dungeon and then your role (DPS, Tank, or Heal). At low levels most specs can perform either function, such as a DPS warrior tanking or a mili healing.
Dungeon Finder can be used to level up all the way to max level.
Areas to quest: Bellshoal (lvl 10-14) Salty Wind (lvl 15-17).
Swamp of Wyrms - Level 19-29
Teleport: Glimmermoor available at level 20.
Riding Skill I available at level 20 (to ride Normal Mounts).
Start quests in Bordhill. This is a good area to use experience blessings because many quests involve killing mobs, and there are plenty of them. Continue questing at Necromancers' Academy, though note that some quests there involve a lot of travel for very little reward. Head to Glimmermoor for more quests. Around level 25, you can quest at Miskatonia and Wyrgast's Library. Port Verecy is one of the last areas to quest, with mobs level 29+.
At level 28, the dungeon Relic's Key becomes available.
The Whispering Islands - Level 29-39
Teleport: Eversummer Port available at level 30.
Port Verecy has a free teleport to Eversummer Port in The Whispering Islands.
Begin questing in Eversummer Port, then Camp of Hemir. Once those are completed, head to Vega Square around level 34~. Around level 36 head to Hall of Reverie or Waker Camp and Buskin Island. Sea Shore will be the last area to quest around level 38.
At level 38, the dungeon Young Deity's Realm becomes available. For this dungeon, it's recommended to have a Guard spec Tank, a Divine Monk to heal, and 2 decent DPS.
The Great Desert - Level 39-48
Teleport: Crimson Tower: Antoria available at level 40.
Riding Skill II available at level 40 (to ride Armored Mounts).
End of Sand and Camp of Sacrathar. Some grinding may be required.
Level 43-46 - Go to Sdukar & do quests that are available then grind. Make sure to get the quests from the big dragon near Sdukar KEY NOTE: Below End of Sand is a group of nomads which give great xp to grind (almost 500xp per person). There are also demons which give 500 xp each (such as burning horns & minotaur invaders & they are found just south of Sdukar).
Level 47-48 - Grind as much as possible & complete the a few quests as well as the daily quests available. KEY NOTE: In Crimson Tower and Sacrathar there are enemies called nobles which again give great xp (almost 500xp per person).
Sinskaald Rift - Level 48-58
Enter the Sinskaald Rift to the town of Stranger's Roost.
Level 52-54 - Complete quests in both Farstrider Outpost and [??]
Level 55-59 - Grind with a ton of daily quests!
Level 60 - Go to Greenmont and there should be a quest near the teleport, talk to the man and he will start chain quests that lead to Under Realm. With the quest, head to Shadow Gate in The Great Desert. Grind Dailies and Repeatable Quests in The Great Desert/Sinskaald Rift.
Under Realm - Level 58-66
There are many quests all over this region. It won't be hard finding quests, but it will be hard getting to level 66 off quests alone.
At 66, head to Eternal Frostlund.
Eternal Frostlund - Level 66-70
Same as Under Realm, plenty of quests all over but the exp required to level from 66-70 is much higher than the previous levels, and the quests alone do not give enough exp to hit level 70. Either grind mobs, do dailies, or use Dungeon Finder when you run out of quests.
Recommended: Slay Mad Nakhit Fisher (Elite) that is located north of Bruto's Camp in Eternal Frostlund. Xp earned is high, but scaled after your level so dont expect to level up once per mob killed.
NOTE: If you've only geared up through quests and not Dungeon drops, the level 70 mobs in Eternal Frostlund will make quick work of you. Best bet is to gear up through Dungeon drops to get better gear if you want to solo the level 70 dailies... Otherwise you'll have to do the Frostlund dailies with a group or team.
Dungeon Finder
For dungeons level 60+, you get 225k exp per run using Dungeon Finder.
58 - MZN (Metal Ziggurat - Normal)
62 - AGN (Abyss of Limitless Gears - Normal)
65 - FCL (Fate of Collapse - Normal) This is the shortest/fastest dungeon for exp.