When you get a BO, it is automatically assigned to his place (if it is empty) and you can use his/her skill in space.
If it is not, I believe it should be used as above mentioned (through assignment screen)
To equip a starship (with looted equipment), you must be in space (haven't tried it while on ground):
open your inventory (key 'i')
Open your starship (i use 'k' then navigate to U.S.S xxxxx and select STATUS on screen; don't know the shortcut yet)
using the mouse, just drag and drop appropriate stuff to your starship slots.
If you want to buy equipment first, go to Sol system, dock, go to shipyard and purchase whatever you can afford.
Equipment in-game can be found (loot), purchased or won through successful mission completion.
