Archaeology is very similar to foraging and botanizing (at least the collecting of fragments). You will need a trowel and a stone chisel and/or metal brush.
You just activate a trowel (at higher skill you can use a shovel for increased difficulty but for now that doesn't matter much), right click a tile (off-deed, on-deed, pavement, dirt, grass, moss, rock.. I believe anywhere except inside mines), and select "Investigate". It will either say "This area looks picked clean." similar to foraging, or you will begin an action timer and start investigating the tile. I generally try to do my archaeology in more abandoned / less developed areas just because I have a better chance of the tiles not being picked clean but you can attempt it anywhere. After the action completes there is a chance that you will find an unidentified fragment and, at higher skill, information about any deeds that were on that land at some point in the past.
I generally try to do my archaeology in more abandoned / less developed areas just because I have a better chance of the tiles not being picked clean but you can attempt it anywhere.
There is apparently a 1/15 chance to get a special anniversary fragment anytime you find a regular fragment. When you do find one, a green message will show up in your event window that says "[17:35:34] You also find another fragment that looks a bit different." and you will have a fragment in your inventory (in a separate stack from the other fragments you have found) called an "Unidentified special fragment".
You can then use your stone chisel and/or metal brush to identify the fragments. Identifying the fragments will slowly remove weight from them (as if you are chiseling away dirt and build-up) and once it reaches the actual weight of the final fragment, it will become identified. For the anniversary fragments, it will be a gift box fragment. Once you find and identify three gift box fragments you can combine them (activate one fragment, right click another) which will create the anniversary gift.
You can also set up a keybind for it. If you use the keybind tab under settings, then "Investigate" is under the Action tab, and "Identify" is under the Item tab.
Hope this helped.