First things first, before thinking about getting equipment for your pet you need to know what exactly your pet can use.
This means find out the pets armor type, to do this pull up the pets Info screen and see what Allowed Armor it can use, second you need to find out what weapon skills the pet has and you can do this via the Minions > Manage screen, select the pet and click skills.
Pets like players can not equip armor they are not allowed to use, nor weapons they do not have the skills for.
So now you know what types of items your pet can use, the next step is to give those items to your pet using the trade system.
Once you have the items in your pets Inventory, go to Minions > Manage, click on the pet then click the Equip screen. You will notice the equip screen is very similar to a player equip screen except there are less item slots to use.
By default the items you traded earlier will be in your pets sell bag so click the red sell bag icon on the top right corner of the Pet Equip screen, this will then load the sell bag contents into the screen.
You should then be able to equip the items your pet has to its body, by choosing slots and clicking the items as you would normally have done when equipping yourself.
Note: When using pets for gathering/crafting purposes you do not need to trade the tools/ingredients to the pet, they should remain in your inventory.