What the best way to use Maugas guns? How about his ult?
His Incediary Chaingun, called Gunny, ignites enemies like Ashe's Dynamite or Torb's Molten Core, after repeated hits. The other, known as Cha-Cha deals critical damage to burning enemies. As you can see, they synergize nicely when you use Gunny first and Cha-Cha as a follow up. You can fire both at the same time or individually, and there are reasons you'd want to do both, but firing both chain-guns simultaneously causes the spread to be very wide. This means you do not want to be doing it at anything past very close range.
If enemies are far off, fire the chain-guns one at a time as they are far more accurate when fired individually. That said, he shreds at close range.
Save Cardiac Overdrive for moments when you are expecting you and your team to take a lot of damage. This is when the enemy is using a lot of ults and you need to try and outlast them.
Practice with his Overrun ability. You can cancel it by pressing the same button, but you won't do the powerful stomp. Also, enemies will roll off you as you run into them. This isn't Reinhardt's charge so you won't suck enemies in. However, if you press the jump button while charging, you decide when to do the big slam that pushes players back.
For Cage Fight, his ult, you may not want to over-commit and trap too many enemies with you in there. Instead, use Cage Fight on isolated targets to secure those tricky kills or to isolate those enemies