You have to assign weapon and ammo upgrades to the weapons via the same screen you choose your weapons and such. There is a button you have to press, but it will tell you what to press when you're in there. As soon as they equipped, that's it then, they work whenever you fire a weapon. They generally improve damage and such.
Yeah, you can pick up weapons and omni gel them later. The Mako is your space buggy that you drive around in. There is tonnes of stuff to find, but the best weapons are ones you can buy (the Spectre weapons) from the guy in the C-Sec department at the Citidel. So selling stuff to get money to buy them helps a lot.