During the main story quest
"The Elf, The Nab-Sack, and the Loom" you'll be introduced to the "Nab-Sack" for capturing beasts, a Vivarium to house them and how caring for them will provide you with materials. At the very end of the quest Deek will give you the Spellcraft for an "Enchanted Loom." The Loom will grant you the ability to upgrade your "Gear" using the materials collected from your cared for beasts.
The materials needed to upgrade the gear varies and the rarer the gear the more materials you'll need to upgrade it to the next stage. For example a "Common" set of Gloves may require 1x Jobberknoll Feather, where as an "Extraordinary" set of Gloves might need 3x Jobberknoll Feathers. Also reach rank of the upgrade tends to require an additional material from another beast. So upgrade one might need 1x Mooncalf Fur but upgrade two will need 2x Mooncalf Fur and 1x Puffskein Fur to upgrade.
As long as you have captured a good variety of beasts and collect their materials regularly it should be pretty straight forward to upgrade an item to level 3. If you wanted to get this achievement out the way, pop a "Common" item into the "Enchanted Loom" and see what beast material is needed. If you don't have said material, head over to "Brood & Peck" the beast shop in Hogsmeade and you can buy the materials needed instead.
Note: Each beast will give you 3x/5x their material every 25 minutes when fed and bushed.