Unlocking weapon slots in Borderlands 2

They're obtained in parts of the main story; one in the early Sanctuary missions and one is unlocked :

when you get back to Sactuary after it takes off. If I remember correctly.

The Relic/Class Mod/Grenade Mod/Shield slots are only locked during the tutorial, they're unlocked extremely early. You never have to do anything but complete story missions to unlock the slots.