(Before you do anything! Download Archy, the Archaeology Assistant addon. It makes your life skilling up this profession ten times easier. http://wow.curse.com/downloads/wow-addons/details/archy.aspx)
If you're a lore nerd/buff/maniac/what have you like me, you'll definitely be interested in collecting all of the neat little trinkets and toys this new profession includes. Each of the major races of Azeroth has something to offer. The profession also offers a few amazing 359 epics, which come in handy with pre-raiding availability. I've included the list of these items here, at the bottom of the post.
But first, I'll cover the main topic; essentially, how to grind Archaeology. First, to make sure you don't go insane, get something else to do while you grind. Movie, TV show, other game, other productive activity, whatever. ~70% of the time you're levelling up Archaeology, you're actually travelling. This leaves a TON of time where you'd just be starting at your flying mount's ass, autoflying across continents.
Now that you're over the fact that you only spend ~30% of your time actually LEVELLING Archaeology, lets get to work. You'll need to go grab your tools from the Archaeology trainer first, one of them is in Garrosh's house. Once you have learned the tradeskill, bind the spell "Survey" (located in your profession tab of your spellbook) to something easily-accessible; if you'll be doing this for awhile, I'd just bind it to 1. This spell is how you actually find stuff. Also located in your profession tab, is the actual profession spell Archaeology. I'd place this on your bars in another easy-to-find location, as opening this gives you a lot of data. It shows your current artifact project and fragment count for each race, and also shows completed artifacts complete with cool lore info for each one.
In a nutshell, Archaeology is fairly simple. If you open your world map up, you'll see 4 digsites on the continent you are on. Starting off, you can only create artifacts from 4/10 of the races: Night Elves, Trolls, Dwarves, and Fossils. However, you can still dig up the fragments of all of the races. If you zoom into a zone map with a digsite, you'll see a highlighted area, basically the same as WoWs ingame questhelper. This area is where your digsite is. Once you're inside the area, use your "Survey" spell. Some surveying equipment will pop up, point towards the general direction of a fragmen find, and light up according to how close you are; green means you're within 40 yards, yellow means you're close, and red means you're far away. Keep surveying until you discover the find and you'll be rewarded with fragments of that race These act as a currency of sorts, applying to the current artifact project you are on. If you go over the artifact's amount of fragments, completing it will apply the spillover fragments to the next artifact, so you never actually waste fragments. Rarely, you'll also loot a Keystone for that race from the find, which is a BoE item that can be used when you are solving an artifact, and counts as 12 fragments towards that race's current artifact project. To use them, you either click the hexagon "gem-socket" in the archaeology book for that artifact, or drag and drop the Keystone into the socket. Each time you clear a digsite (three finds per digsite) another one is randomly opened on your continent. You get 3-6 fragments per find (9-18 per dig site).
Now for the more tricky/annoying stuff. Each of the races have a TON of common grey artifacts that really serve no other purpose than to give you a little lore info and be vendored. There are also "tiers" of artifacts, level ranges of Archaeology in which you are able to discover more artifacts. Many of the best artifacts can only be discovered at 300+. Every time you complete a project and start a new one, you're automatically given a random artifact from the race, the common ones obviously being more common than the rares or epics. Before any duplicate common artifacts are found, you must complete all of the artifacts currently available to you, which is probably the one GOOD thing about levelling this profession. Rares and epics are just that, rare. You have a small chance after completing any artifact to "proc" a rare or epic.