Hi,
you get a second slot at level 20 by doing a mission from the Icon vendor in Steel Canyon ( I think, not sure ).
I don't remember the details for the other slots but the mission comes from Icon vendor.
Founders Falls and Independance Ports for the others.
Yes as mentioned you do a mission for the tailor at lvl 20 for an extra slot, at 30 and 40 making 4 in total including your original.
You can also obtain certain salvage items (available during an event or through WW) and exchange them for a 5th slot.
you can also do extra missons for extra costume bits
lvl 20 - cape misson from the Atlas city rep
lvl 30 - auras from .... ithink Atlas city rep
you also get extra costume bits as Veteran rewards e.g. samurai armour around the 18 month mark
Also you can get a bonus slot by collecting Halloween salvage during the event which starts on Monday 
or you could just buy the salvage from teh auction house, there are 4 pieces if i remember and they are very cheap.
You then take them to a contact in Croatoa
How do i get a 2nd, 3rd costume so that i can change from time to time