Alliance Level 6 to 12

You should be starting these levels at the Inn in Goldshire. In Goldshire, pick up the quests Rest an Relaxation, Kobold Candles, Gold Dust Exchange, A Fishy Peril and The Fargodeep Mine. Completing these will give you 2 more follow up quests.

The area around Goldshire can get pretty crowded and you may find competition for the mobs needed for the quests, especially the Kobolds. If there is already a group at the mines, ask politely if you can join them. Although this won't give you as much experience it does make completion of the quests faster and easier.

Sometimes it's easier to just grind out a couple of levels around 7 to 8. The area is thickly populated with mobs, so you can just run in any random direction, killing everything you see. Kill humanoids whenever you can and save the linen cloth they drop.

There are more quests at Stonefield farm to the south of Goldshire.

If you follow the questing sequence from Goldshire you will be around level 8 or 9 by the time you get the quest Shipment to Stormwind. This leads you to the nearest major city with an Auction House and profession trainers. Catch up on your class skills in Stormwind. Find the profession trainer for First Aid. Any guard will give you directions and a convenient marker on your mini-map. Train in first aid and use up all that linen cloth you have been saving by making bandages.

Select skinning and one of the other gathering professions (mining or herbalism) for your first 20 levels. Sell anything you gather in the AH. The gathering professions will make you enough gold for decent equipment at level 20 when it starts to make a difference.

After training, head back to Goldshire and finish any outstanding quests.

Around level 9 or 10, I usually head over the bridge into Westfall. There are two quests right on the path after you cross into the zone. These and a little grinding on the mechanicals will easily get you to level 12.