TheBlastMage answered about how to save and equip gear sets, so I'll just address the second question.

Quote Originally Posted by Pikachu_Pizza View Post
Do you have to carry around both gear sets? Or does each class have a different bag when you switch?
Yes, you need to carry the gear for all your gear sets in order to be able to switch between them. Your Armoury Chest can hold up to 25 items of each gear type (25 weapons/tools, 25 hats/helmets, 25 chest pieces, etc.) When you switch between gear sets, that's where the system will look for all the pieces you need for the gear set you're switching to.

If you're missing the main hand weapon or tool that was recorded for that set, it will search your Armoury Chest for another weapon/tool for the same class and ask if you want to substitute that one. If you don't have any weapon/tool for that gear set's class/job in your Armoury Chest then it will fail to load that gear set at all. If you're missing any of the other pieces of the set other than the main hand weapon/tool, it will swap out the rest of the set but nothing for that slot. Oh, and if the set you're switching to has any empty slots (either designed that way or because of missing items) so that switching involves un-equipping more items than you're equipping, you have to have enough space in your Armoury Chest to hold the extra items.