This is the one profession most people skip and try to purchase things in the Auction House, unfortunately there is no way to really level mining using Auction House Supplies except that you can smelt your way up almost halfway through but it gets expensive.
Since patch 2.4 was implemented you can now train up to Artisan Mining at any trainer available. To get your skill past 300 you will need to see the Master Mining Trainer that is located in each factions appropriate starting city for the Burning Crusade in Hellfire Peninsula, Honor Hold and Thrallmar.
This first step is fairly easy and there is a Horde and Alliance area included on the map. The target zones are areas that are the thickest in mines, meaning if you stood in the center of that circle you will probably always see at least one node up, unless it was recently mined. These are not the only places for mines in these zones they can be found all over the zone along mountain ridges and near water, try making your own paths between the circles to level your skill effectively. Other zones of similar level range also contain these nodes along mountain edges, in caves, and near water.
This part of the mining guide includes the mining of Tin Veins and Mithril Deposits. The circles represent the most dense node areas and it is recommended you create your own path to go between the circles in order to maximize your mining efficiency. You may also encounter Silver Veins and Gold Deposits, these are rarer nodes and aren't a reliable leveling source but if you find them go ahead and grab them. Other zones of similar level range also contain these nodes along mountain edges, in caves, and near water.
This section of the mining guide will deal with having to go in a lot of caves, although you can run around the circle areas on the map and find above-ground mining nodes. In Tanaris you can complete an entire circle of the outskirt of the zone along the mountains and pretty much the same for The Hinterlands, especially if you don't want to drop in the caves to mine. Other zones of similar level range also contain these nodes along mountain edges, in caves, and near water.
You may encounter Truesilver in this part of the guide, it is a rare node and if you find one go ahead and get it. In this section you will find both Rich and Small Thorium Veins in all 4 of these zones as you are looking for nodes. The circles indicate the best spot for the two different types but be aware that you will be able to still level off Small Thorium Veins you see while looking for Rich Thorium Veins. Other zones of similar level range also contain these nodes along mountain edges, in caves, and near water.
If you refuse to go to Outland there are still skillups to be had from Thorium Veins but if you want to do this fast then you definitely should head to Hellfire Peninsula. Hellfire Peninsula is so densely populated with Fel Iron Nodes that all of these circles don't even do it justice. You can pretty much make any path around the mountains you wish and find enough Fel Iron to level effectively. Zangarmarsh is an alternative to Hellfire Peninsula but if you should only do Zangarmarsh if you are extremely bored because Hellfire Peninsula has a lot more nodes.
Some of the Adamantite Deposits will require you to have a flying mount; most notably, the northwest and south east circle in Terokkar Forest and the Northwest area in Nagrand. You will also probably encounter Khorium in your travels, make sure you gather all these. You don't really have to mine Adamantite to get to 375 you can keep doing Fel Iron but you may enjoy the break and speed of getting Adamantite.