How do you unlock more bank slots?

I’ve only got the top two rows to store stuff…granted I’m only level 13, but its these class mods I’ve got. They’re taking up six or so inventory slots because the bank is already full of them.
I’m not sure how they work…should I hold on to all mods applicable to my class? Can you pay to unlock more bank slots?

Pardon if these are some basic questions, Google has nothing for them as far as I can tell.

buy sdu upgrades from Marcus on sanctuary .