When you first start out the majority of your exp will come from quests. So try to fill your log with as many quests as you can. But don’t always take the ones that require you to visit a dungeon or kill an elite. These take organization so you won’t get to them right away and they’ll just sit in your quest log taking up space for other quests. Around halfway to level sixty though quests begin to slack off in terms of experience and you will be forced to do a lot of grinding.
When grinding, or killing mobs repeatedly for exp, always pick your targets carefully. One of the best things to do is try to pick mobs that give some type of reputation gains when killed. You generally don’t see these until higher levels, but that’s when you’ll be doing most of your grinding anyway. Another good idea is to grind on mobs that bring in the most cash. Humanoids drop money and cloth and have a slightly higher chance of dropping green armor and weapons. Beasts will only drop animal parts that don’t sell for much. But if you are a skinner you can make money that way.
You must also use the Rest System to your advantage. It can double the amount of EXP you get so always try to log out at an inn. When I got to level fifty with my paladin I typically let him rest up till he had a full health bar and went and leveled up some of my lower level alts in the mean time. It took more days to reach sixty, but I spent fewer hours grinding.
Finaly, many of the areas that I advise you to level in are hard to find unless you have been there before. Here is an atlas stie to help you find your way around.

Alliance Leveling
1-5
If you are a dwarf you’ll start out at Coldridge Vally in Dun Morog. Get your quests from the locals and explore till you reach level five. Now head north east and travel through the tunnel to reach Kharnos.

If you are a Night Elf you will start out in the Shadow Glen of Teldrassil. Talk to the citizens nearby to load up on quests and when you reach level three head down to the giant tree in the center of the area to load up on more.
If you are a Draenei you will start out by waking up from an escape pod in Azuremyst Isle. Head down to the main wreckage of the ship at Amman Vale and load up on quests. At about level six or when you have run out of quests head west till you find a road leading to the main part of the island.
5-10
If you are a dwarf load up on quests from the people inside the inn and head out. Dun Morogh is more spread out so if you want to complete all your quests you will need to go to Steelgrille Depot in the north, Brewnall Village in the west and the two guard towers in the east.
If you are a Night Elf finish up any quests you have in Shadow Glen and head south down the road and then west to Dolanar. Set your Hearthstone at the inn and load up on quests. There are some quest givers hidden in the other buildings so check everywhere.
If you are a Draenei follow the Road until you get to Azure Watch. Set your Hearthstone here and accept the available quests. When you reach level seven head down to Odesyus Landing in the south and take the quests here. When those are done head back to Azure Watch and get new quests that will take you to level ten. At this point you will have a quest that takes you to Stilpine Hold which will be your next big quest hub.
10-15
If you are a Dwarf go through the southern eastern guard tower and into Lock Modan. Because this zone is thin and long it is more efficient to finish all the quests in the bottom half first. Be sure to check inside the guard towers as well since there are a few quest givers hidden inside. Once you have finished up here head north along the road until you reach Theslmar.

If you are a Night Elf Finish up any quests you have in Teldrassil and dead to Auberdine. To get there head through Darnasus and then left past the bank until you see a giant tree with a purple glow at its bottom. Walk through it to be transportation

to Ru'ruthan Village. You can then take the boat to Auberdine or hop a free hipogryph. Once there reset your hearthstone at the inn and load up on quests.
If you are a Draenei finish up the quests in Stillpine Hold and when those are done head north from Azuremyst to Bloodmyst. Go through Kessels Crossing and head north to Blood Watch. Reset your Hearthstone and fill up on quests here till level 15.
15-20
If you are a dwarf take all the quests you can in Theslmar and head out to complete them. You will also want to head to the north guard tower and the hunting lodge in the south east for a few more. At about level eighteen the loch will run dry on quests so take the tram from Ironforge to Stormwind.
If you are a Night Elf you will get nearly all of your quests here in Auberdine. There are a few though that require you to run to other parts of Darkshore like the Buzzboxs scattered over the map and the grove of the ancients in the south. When you have reached level twenty hop the boat to Menethel Harbor. Follow the road east to Loch Modan and keep following it to Ironforge where you can ride the tram to Stormwind.
If you are a Draenei keep grinding on the quests from Blood Watch until about level 18. Nearly all of them will be given in Blood Watch, but there are a few on Wyrmscar Island in the North East. When you have done those head to Auberdine and take the ship there to Menethel Harbor. Run east along the road and follow it to Loch Modan. Keep following the road until you reach Ironforge and take the tram to Stormwind.
20-25
From this point on all the alliance races should follow the same leveling path. Start by leaving Stormwind and heading east to the Redridge Mountains. Load up on any yellow quests from the people of Lakeshire and level here until 22. Then head west and south into Duskwood until you reach Darkshire. Take any yellow quests here and alternate between Duskwood and Lakeshire until level 24.

25-30
Take any new quests from the people in Darkshire and head off to Raven Hill. Since this is on the other side of the map you can get there more quickly by flying to Sentinel Hill and heading east across the river. Many of the quests you got in Darkshire are completed here so check back frequently for follow ups. At level 25 fly to Menethel Harbor and take any yellow quests here until 27. Eventually head back to Darkshire and finish off any quests you have here until level 29. Now go to Ashenvale in Kalimdor and take all the quests in Forest Song, the alliance camp in the north eastern half of the zone.
30-35
At level thirty head south from Duskwood into Stranglethorn Vale. You will have three major quest hubs here. The first is a rebel camp just to the right as you enter from the north. The second is Nessingwary's Saffari which is further south and off the road to the right. And your final hub will be Booty Bay all the way to the south. To reach it swim along the coast instead of trekking though the jungle, otherwise you'll get attacked by monsters. Start off by taking the quests from the rebel leaders. When you have done all the yellow ones head down to Nessingwary's Saffari and take the first hunting quests. Don't bother with the Hills of Stranglethorn quest though. When you are finished you should be close to 33 or 34. Head north to the Hilsbrad Foothills and complete the quests in Southshore. Most of them should be yellow to you by now, especially the ones in the Alterac Mountains.
35-40
AT this point you should consider heading to Dustwallow Marsh. Most of the quests will be yellow by now and easily completed, and there are some good quest rewards too. When you only have orange quests left head back to Stranglethorn and head to booty bay. Search around the inn and the docks for several quests and work on them till level 38 At this point head north again and take the quests at the refuge camp in the Arathi Highlands until level forty.
40-45
Get your regular mount and head back to Booty Bay. You should finish up any remaining quests by level 42. Most of them will be in the southern half of STV. When you are done hop a flight to Menethel and take the boat to Theramore. Swim along the eastern coast until you reach Tanaris and

Steamwhedle Port. Head West along the road until you reach Gadgetzan and base yourself here. Take all the quests and when you run out travel to Feathermoon stronghold in Ferales.
45-50
Finish up any quests in Feathermoon and head back to Tanaris. Take the quests from Steamwheedle Port and go south to the pirate base. This is an excellent place to grind because the pirates drop footlockers that may have a key to a treasure chest inside one the of the ships. The chest seems to have a high chance of dropping a blue so you can make some good money here. When you grow tired of this place head back to Azeroth and travel to the Hinterland. Do the quests from the Wildhammer Dwarfs until you reach forty eight. At this point head back to Kalimdor and head up to Felwood through Ashenvale. Stop when you find the Emerald Circle base camp. A little off to the left is a camp of Deadwood Furbolgs. Killing these creatures gives Timbermaw Hold reputation which is necessary to gain access to the tunnel to Moonglade and Winterspring. You will only have to kill enough to get to unfriendly with them before they will let you through the tunnel and by then you should have reached level forty nine. At this point go back to the Emerald Circles camp and take any quests they may have. This should get you close to fifty. Finish off your level by flying south and loading up on the quests at Marshals Refuge in northern Ungoro Crater.
50-55
Keep working on the quests here in Ungoro until level 52. Give one last wave to Kalimdor and head to northern Azeroth. Start by heading to the Western Plaguelands and basing yourself at Chilwind Camp. There will only be a few yellow quests at this point so when you reach 53 or are close to 54 head to Winterspring and Everlook. Do most of the yellow quests here but don't bother after level 55.
55-58

Return to the plauge lands and finish up any quests at Chilwind point. Each of the farms in the northern half of the zone also have some quest givers so make sure to check them carefully. When you are done you should be near or just past level 56. Take the road east and head towards the Eastern Plaguelands. Keep following the road until you near an abandoned town Called Corwin’s Crossing. Go around it and keep heading east and north until you find Lights

Hope Chapel. After you have established a base there load up on quests and head out to finish them. If you reach level fifty seven and have no more quests you can head back to winterspring and get a few more from everlook. When you reach fifty eight head for Outland.