You won't gain any real end-game benefits from Archaeology, most of the items you can get have no in-game power like pets, toys and mounts. This profession is mostly for fun. However, most of the armors and weapons you can get are bind to account.
Once you train Archaeology, you will be able to see dig sites on your World Map. They look like little shovels. There are always 4 dig sites per continent (only 3 for the Broken Isle), and it won't change on logout, server restart or anything. You have to dig one out to get a new dig site. The number of zones and continents you can find dig sites depends on your Archaeology skill and your character level.
You will also get a new ability called [Survey]. You can only use this ability when you are inside of a dig site. The Survey tool will spawn and point in the approximate direction of a fragment, and there is a light that will indicate the distance you are from a fragment.
When you discover your find, you'll get fragments specific to a particular race. There are 10 races, it depends on the zone, and your Archaeology skill which one you will find. Each dig site can be searched 6 times before it despawns and a new site spawns. (8 times at Draenor).
When you loot your first fragment, you will start a research project. The number of fragments required for each artifact can vary from 25 to 185. When you have enough fragments, you can click the little Solve button to complete that Artifact. If you have excess fragments, it won't go to waste. They will start a new project. Unfortunately, you cannot control which research projects you get, it's random, but it's based on your Archaeology skill level.
You will get skill points for surveying dig sites, and 5 skill points for any artifact you solve (15 for rare ones), no matter which race it’s for. For example, you can level up to 600 in Eastern Kingdoms alone.But, after you reach 600, you can only gain skill points if you go to Draenor, Broken Isle, Kul Tiras or Zandalar and after you reach 700, you can only gain skill points in the Legion zones at the Broken Isle, or the new Battle for Azeroth Zones, Kul Tiras and Zandalar. For the last 150 points between 800-950, you will only gain skill points in Zandalar or Kul Tiras zones. If you have an epic flying mount and you level in zones where you can use flying, then it takes about 10 hours to level up Archaeology to 950.