Below is the list of quests I have been using to level from 6 (around which you should be leaving the start area) to 10 (the upper level range for Elwynn).

Rundown of the Quests
Below is a quick rundown of what each quest involves, where it will take you and the mobs you’ll be grinding.
Batch 1

These three quests will take you South of Goldshire and into the Fargodeep Mine. You’ll be grinding level 6-7 Kobolds here and it shouldn’t be too hard to do, dependant on your class you will breeze through this or do so at a moderate pace without any problems.
When you’ve been through the mine you should have the majority of the [Large Candle]s and [Gold Dust] required. Grinding the mob outside the mine will fill the rest of the quota. From here it’s back to Goldshire to hand in.
Batch 2
Once you hand in the candles quest you’ll get the Shipment to Stomwind one, I always do this straight away as you’re right next to Stormwind and it gives a good buff for the rest of the questing, I always take [Oil of Olaf] for the armour buff.
Batch 3 + 4

Hand in the Fishy Peril quest to Dughan, collect the rest of the quests and head off down the road. The Jasperlode Mine is another easy quest if you take your time. You’re probably about level 7 now and will be facing level 7-8 mobs, including the wandering bears and wolves. My tip is to head straight to the Tower of Azora and head North from there. You can get away with only a few encounters.

Actually doing the quest is easy, get about half way through the mine and you’ll see your message saying it’s complete, then just wander back out. This should take no longer than 5 - 10 minutes meaning there’ll be no respawns on the way back if you’re lucky. Don’t hand this quest in yet — it’s a long walk back and forth to Goldshire.
Batch 5
You’ll have collected this quest when you handed in the Fishy Peril one, it’s simply a quest to talk to Guard Thomas. Just follow the road from the Tower of Azora to the river where Thomas is standing.
Batch 6
I’d recommend going to the logging camp and picking up the two quests there before setting off on this one as they can somewhat overlap and it saves walking back again.
Just head North up the river for this one, you can avoid any wandering wolves easily, if not it shouldn’t be anything you can’t handle.
Find the mangled corpse and hand in the quest, accepting the next part at the same time.
Batch 7

This one is more tricky, you’ll have to swim up the river and walk round behind the murloc camp, it saves getting ganked. At this stage you’re probably a high 7 or low level 8. You will be taking on mobs of level 8-10 and in some cases they pull in twos.
Taking a friend along for this one can help if you’re one of the low armour or health classes. You’ll need to carefully pull the murlocs away from Rolf’s corpse, especially watching out for wandering murlocs or pulling by accident due to fleeing enemies.
Once you’ve cleared the two murlocs next to the corpse you should be able to hand in the quest and accept the next one. Whilst you are around this area you may see some wood piles which are for A Bundle of Trouble, pick up any if you can.
Batch 9

This can be a long drawn out affair. Killing the prowlers and bears is simple enough if you take your time and heal up, bearing in mind that these beasts are level 9-10 and you’re probably 8-9.
Finding the wood is the main problem as it is scarce and a lot of people seem to do this quest making it even scarcer. Doing this in a group could land you grinding there for half an hour, but that’s all good XP.
Once you’ve filled up on wood and killed all the beasts you can head off to hand in the last quests we’ve done.
Batch 10 + 11

Head back to Goldshire, hand in Deliver Thomas’ Report and get the Westbrook Garrison one.
Head along to Westbrook and hand in, you can then get the Riverpaw Bounty quest and the Hogger ellite quest.
Batch 12

You’ll be grinding level 9-10 Gnolls here, some of which sit in camps of 3-4. Doing the bounty quest is soloable if you stick to solo targets, i.e. stay away from the camps. Doing Hogger however is tough as he is a level 11 elite which probably puts him 1-2 levels above you and also due to the fact that he is an elite spawn.

Getting help with this is preferable, either a few other same level players or a higher level player who is passing by. Grinding the mobs can be useful before you go for Hogger as it reduces the chance of adds being pulled which is not something you want if you can avoid it.
You can now hand in the bounty quest at Westbrook and head to Goldshire to hand in the Hogger one.
Batch 13
You may have pinged 10 now but there’s a few quests left if you haven’t, or if you want to complete the entire Dughun/Thomas chain.

We head off towards the logging camp to hand in Cloth and Leather Armour that you will have gotten from Dughun. On the way take a right off the road to kill some Defias members to get [Red Linen Bandana]s for the Red Linen Goods quest. You may have to grind around Blackwell Pumpkin Patch if you’re not getting the drops.

Once you’ve handed in these quests you should be level 10 or nearly there. If you still need XP it should be easily grindable on the level 9-10 mobs around the logging camp.
I always do the level 10 class quest straight after doing these quests and try to upgrade my weapons and items at the same time, it is quite a turning point in the game, that is all the class specific stuff starts to come into play like pets and talents.
Anyway, that’s about it for the initial push, I always tend to level in Elwynn as it seems more tightly packed than Dun Morogh or Teldrassil.