Staff and Hand Gathering
You can unlock the rank 2 of these as soon as you unlock the Abilities menu at level 25. These are bonus skills awarded for spending a small amount of points however you like.
Rank 3 has the same method, but it takes much longer. You gain AP through various methods, including finding landmarks, tasks you're asked to do, and leveling.
Other Tools
Rank 2 gathering on all tools is available as soon as you get the tool; you need to max the first effect row of each one to get the "Master" effect, such as Master Cutting.
Rank 3 gathering is locked to the higher tiers of these items, and you will unlock them much later in the game. Furthermore, you will need to use the restricted panel (with the holes in the board) to be allowed to get the maxed effect. The general process is to create as many links as possible. Doing character events will give you some skills to help with that, and some catalysts are helpful as well. In particular, "Components Divided" is easy to use on catalysts as it allows you to place components over each other without deletion. You want at least a 6x6 board as well.
For unlocking each of the tools, you are required to do Major Gathering (minigames) a set number of times. Here are some locations you can enter and exit out of quickly:
- Sickle: Windswept Plains > Newborn Verdure
- Hammer: Quartz Lapis > Quartz Terrain (there are a bunch within a short distance of each other)
- Fishing Rod: Flower Bank Ruins > Sea King's Domain > Northwest
- Bug Net: Near-Forest > Near-Forest > Southwest
- Pickaxe: Neil's Shore
- Slingshot: Quartz Lapis > Quartz Terrain