Archaeology items are quite rare drops that you can get either by fishing or digging on a 'Dig Spot'. There are a total of 10 different items, 5 obtained by fishing and 5 by digging. What you get also depends on the biome you're currently on, so each biome always give you one specific item when fishing or digging. For a better break down look at image below:

Fishing is done by building a fish trap on a water tile and then wait and pray that it catches you that damned Skeleton Fish. It is quite rare that it catches an Archaeology item but there are a few ways to improve the speed but it seems it's all up to RNGod to bless you:

• Fishing skill point: fishing trap catches things 50% faster
  
• Liquid luck potion: increase rare items drop chance and double items received
  
• Lighthouse: items gathered are increased by 50% and this effect can be stacked, within radius
  
• Hoarder shrine buff: increase number of items dropped
  
• Industrious shrine buff: all structure work faster and it looks like fish traps are also affected
  
• Builder scroll: boost structure work speed within radius, doesn't look like this buff can be stacked

Drone are an essential quality of life when fishing and can be used to AFK fish farming because they automatically collect items fished so you don't have to interact with each fish trap one by one...

Dig Spots are naturally occurring spots that appear on empty land and can be dug using a shovel. They generally drop bone, sand, arrow, coal, gem on any land but have a rare chance to drop an Archaeology item depending on the biome it spawned.
Waiting for the them to naturally spawn is not a doable strategy unless you really want to kill some time by hunting Dig Spots. Hence shrines are a required for the Excavator buff which fills the land they are currently placed on with Dig Spots. So you'll need to spread shrines one on each land and across all biomes to get all 5 items.

Archaeology items are used to fill the Museum's Archaeology Bundle section which requires one of each type to be completed.
They are also required for end game crafting in order to upgrade rods to their final form. Rods can be upgraded at the Spirit Crystal and require the following materials:

• Meteor Rod: 10x Lava Eel, 10x Fossil, 10x Star Fragment
  
• Storm Rod: 10x Anchor, 10x Sunken Ankh, 10x Star Fragment
  
• Blizzard Rod: 10x Frozen Relic, 10x Frozen Squid, 10x Star Fragment
  
• Death Rod: 10x Skeleton Fish, 10x Sphynx, 10x Star Fragment