For the fastest possible archaeology research, it's preffered to be a mage (or have a mage friend willing to tag along) and have Azeroth flying with 280%+ flying speed.
For the impatient read, i have boiled the guide down to one word:

Without patience, you won't get any neat toys, such as mounts and pets.

Now for the real guide (the one for the patient):
First of all, read the guide in your archaeology book, it describes the basics of how the proffesion works. Then choose one continent, either Kalimdor or Eastern Kingdoms, depending on which kind of artifacts you want to uncover. Kalimdor is mainly Nightelven artifacts, while Eastern kingdoms is a more balanced continent, but with dwarwen and troll artifacts as a main focus.
I won't spoil the fun of telling what you get and where, if you want to know that, use wowhead or thottbot.
After that just fly from digsite to digsite and collect the artifact fragments. Do NOT assemble any before skill 100, as you stop gaining skills for collecting fragments, at this skill level.
At skill 300 you may go and find artifacts in Outland, Orc and Dreanei, at skill 375 Northrend, Vrykul and Nerubian, and last about skill 450, you may start seeing digsites in Uldum, these are a part of the Kalimdor spawns (there are always 4 "dig zones" marked with a shovel on your world map, each containing 3 digs). Uldum gives Tol'vir artifacts.
With this knowledge the final words of wisdom are: Be patient.