Not sure if the tutorial explains this, but make sure you pledge to different contacts to unlock new weapons and upgrades. Press J and go to the progression tab to see the different contacts available for each faction. Double click on one (that's in your district) and you can see what unlocks they offer and pledge to them.
Each side has 2 factions, each faction has 2 upgrade paths. One unlocks vehicles and vehicle mods, the other unlocks weapons and character mods. Reaching milestone numbers of kills with specific weapon classes unlocks weapon mods and upgraded versions of those weapons, you can see these on the roles tab.