There are three guns that are fairly simple to upgrade simply because it does not take a lot to max them out.
Tranquilizer Rifle: You can have this in your hand as you start the first mission as long as you tell Sarif you want to take people out from a distance without killing them. To max this baby out you need:
1x Target Leading System
3x Reload speed upgrade
Double Barreled Shotgun: Slightly more difficult to get UNLESS you have the Tactical Enhancement DLC. If you don't, then you have to wait until your 2nd trip to Hengsha to snag this one.
To max it out you need:
1x Damage output upgrade
2x Reload speed upgrade
Rocket Launcher: You can snag this one as early as Derelict Row, and it is EXTREMELY USEFUL on bosses if you don't upgrade your typhoon
. All it takes to make this one uber-epic is:
1x Heat targeting system
4x Reload speed upgrade
You can find most upgrades during missions OR buy them from merchants as the game progresses.