First you train it and now look at your map
http://img59.imageshack.us/i/wowscrn...010035125.jpg/
The shovels show where you can find the artifacts. Now that you know where to go its time to head on over there.
You finally arrived at the place, now you get to use your nifty new ability "Survey" Its a 1 second cast with a 4 second cooldown. What it does is plant a thing that looks like this on the ground.
http://img237.imageshack.us/i/wowscr...010035555.jpg/
The little bulb shows how far you are from the nearest artifact, red is very far yellow is far and green is close. Now to see which way you go you look at the direction the scope is pointing and head in the direction.
Check every once in a while for progress and to make sure you're still going in the right direction. Here you find out you are closing on the artifact.
http://img6.imageshack.us/i/wowscrns...010035731.jpg/
You are very close now, you should use the Survey ability every few steps.
http://img508.imageshack.us/i/wowscr...010035900.jpg/
Now you just keep using Survey every few seconds until something like this pops up instead of the scope.
http://img39.imageshack.us/i/wowscrn...010040018.jpg/
Now for the best part! LOOTING!
http://img205.imageshack.us/i/wowscr...010040058.jpg/
Here are screenshots of other stuff that has to do with Archaeology.
http://img812.imageshack.us/i/wowscr...010040224.jpg/
http://img824.imageshack.us/i/wowscr...010040235.jpg/
http://img710.imageshack.us/i/wowscr...010040239.jpg/
------UPDATE 2------
The things you loot from whatever you find do not take up inventory space, instead they work kinda like emblems but instead of your currency page you keep track of the stuff you looted in what I like you call, your Archaeology journal.
It seems as you collect the artifacts of a specific race you have a bar to fill
http://img825.imageshack.us/i/wowscr...010042145.jpg/
For this I needed 30 of them to "Solve" the mystery or something like that.
So you click solve and it combines 30 of your pieces into a new item, now because my level was low I obviously didn't get anything amazing. Just vendor trash, but it also leveled my skill by 5! Drop rates currently seem to be 2-4 per find.
http://img203.imageshack.us/i/wowscr...010042218.jpg/
Also for those of you wondering how to tell if you're in the zone when you get closer, it shows a highlighted area like you would see in quests. http://img822.imageshack.us/i/wowscr...010044621.jpg/
Currently you cant just pick one place and grind it, after some time you will run out of things to find and have to move on to another location. A great way to encourage people to explore. But don't worry they respawn and will show you on your map, so you will always know when you can go back to that zone.
Feel free to ask questions, I may not know a lot about it right now. But I will answer them as best as I can.
And because many of you are probably going to ask, yes there are shovel icons in Northrend and Outlands.
Current list of what area gives what that I know of
Night Elf Archaeology Fragment:
Ashenvale, Stonetalons, Feralas, Hyjal, Felwood
Tol'vir Archaeology Fragment:
Uldum
Fossil Archaeology Fragment:
Un'goro, Desolace, Dustwallow Marsh, Southern Barrens, Eastern Plaguelands, Redridge
And what would Archaeology be without some Dwarf artifacts to find?
Dwarf Archaeology Fragment:
Hillsbrad, Wetlands, Twilight Highlands(It took ninja skillz to do that as an 82), Searing Gorge
Troll Archaeology Fragment:
Stranglethorn
Orc Archaeology Fragment:
Hellfire Peninsula, Shadowmoon Valley, Nagrand
Draenei Archaeology Fragment:
Shadowmoon Valley, Netherstorm, Zangarmarsh
Current things I've solved(completed). Keep in mind that my level is low so they're greys to vendor but they have some nice text on your little journal.
Nelf:http://img267.imageshack.us/i/wowscr...010044438.jpg/
http://img835.imageshack.us/i/wowscr...010054837.jpg/
Fossil:http://img835.imageshack.us/i/wowscr...010052136.jpg/
http://img64.imageshack.us/i/wowscrn...010063118.jpg/
Dwarf: http://img214.imageshack.us/i/wowscr...010065857.jpg/
----How Rewards Work(My theory atm)----
It seems that what you create is already decided before you actually create it, but what it picks for you is random. To create it you collect the number of fragments it asks for(ranging from the 20s-30s for common ones and those who got lucky and got the pet version said they needed 85 fragments) So you will know when you finally struck gold and can create a rare artifact, which would then make you go the extra mile.
-----UPDATE 3-----
Tablets, what do they do and where can I find them?
http://img820.imageshack.us/i/wowscr...010071332.jpg/
The Orc version is called "Orc Blood Text"
The Draenei version is called "Draenei Tome"
Thanks to others for finding these next ones
[Highborne Scroll] Found searching within the Nighelf bowl
[Nerubian Obelisk] Found searching the Nerbian bowl thingy
And finding out how they work for us, we have pinkshirtbadman to thank for that.
----How leveling it up works----
At first you gain levels by digging the stuff out, but that isn't the main way to level it(well isn't currently). Imagine that part like disenchanting. After some levels it stops giving them.
The way you level it from what I have seen is by completing an item with the required number of fragments. You can currently dig up anything as a level 1 Archaeologist. But you can't complete the items until you get the appropriate rank for that specific type of fragment. The ranks are when you learn the next step of the profession, like Journeyman etc. You will unlock a new set in your journal with each rank.
---Achievements---
http://img713.imageshack.us/f/wowscr...010080502.jpg/
http://img819.imageshack.us/i/wowscr...010080512.jpg/
http://img411.imageshack.us/f/wowscr...010080526.jpg/
And as a reminder the dig you yourself go on is only for you, don't expect to be competing with others to find it first. The pvp part was because it would gather others to the same site at times, and we all know what happens when others see a red name on their screen