Aside from getting the 4-slot bag from the quest line, you have to either buy AP to get 5-slot bags or buy them off of people who donate for AP in exchange for gold.
As well, sometimes the GMs will host events where they'll give away bags, usually 7-slot bags, so I'd pay attention to event rewards because it could mean a chance at one of those.
The blueprints for some of the bags often require 5-slot bags and the largest bag before it, which again, all require 5-slot bags. In short, the "bag system" ultimately revolves around 5-slot bags.
Rarely, they'll have some of the larger bags (10,15,20 slots) in tiered spender events. But those are the only ways you can really get a bag. AP or giving gold to an AP buyer who buys it for you (or sells it on the AH if you prefer buying it there).