I will do my best to explain this for ease of use and getting all weapons unlocked.
On the weapon upgrades screen at the Mothership you will have 3 square coloured boxes which you use to select your weapon type. You will need to combine different types to unlock them and it will cost a fair amount of moolah. So your best bet is to wait until you have a few trillion.
Below is the list and in what order from top to bottom how the boxes should look.
Mine Field:
Blue Laser Box
Empty Box
Blue Laser Box
Rubberised Kinnect Flak Cannon:
Red Rocket Box
Empty Box
Blue Laser Box
Particle Beam Refractor:
Green Bullet Box
Empty Box
Blue Laser Box
Machine Gun:
Green Bullet Box
Empty Box
Empty Box
Ion Accelerator
Blue Laser Box
Empty Box
Empty Box
Missile Barrage:
Red Rocket Box
Empty Box
Empty Box
Wide Shot:
Green Bullet Box
Green Bullet Box
Empty Box
Love Gun:
Green Bullet Box
Red Rocket Box
Empty Box
Mega Missile Volley:
Red Rocket Box
Red Rocket Box
Empty Box
Blast Barrage:
Red Rocket Box X3
The Obliterator:
Green Bullet Box X3
Super Mega Ultra Laser:
Blue Laser Box X3
Energy Arc:
Green Bullet Box
Green Bullet Box
Blue Laser Box
Gravitation Aperture Gun:
Green Bullet Box
Green Bullet Box
Red Rocket Box
Targeted Photon Beam:
Green Bullet Box
Blue Laser Box
Blue Laser Box
Laser Unicorn Attack Squad:
Green Bullet Box
Blue Laser Box
Red Rocket Box
Plasma Saber:
Green Bullet Box
Red Rocket Box
Red Rocket Box
Plasma Hurler:
Blue Laser Box
Red Rocket Box
Red Rocket Box
Singularity Pockets:
Blue Laser Box
Blue Laser Box
Red Rocket Box
I believe this is all correct for the full 19 weapons available to you. Like I said, it takes a lot of money to just buy them, let alone upgrade. Upgrades will prove invaluable if you wish to devastate your enemies.