When Frank is a distance from Adam, Adam will blow a balloon full of poison gas.
Adam will spin his chainsaw.
Adam will slice Frank with his chainsaw.
When Frank is near Adam will blow fire.
When Frank is a reasonable distance from Adam, he will throw knives.
At the start of the fight, Adam is on the far side of the Space Ride track. Head shots are effective against him, but the best tactic is to run out of the Space Ride station away from Adam and onto the second floor walkways around Wonderland Plaza. His dual small chainsaws are capable of doing a lot of damage, so Frank needs to keep a distance and shoot the balloons he inflates before he throws them at Frank; his balloons will damage him much more than almost any of your weapons can, with the bonus of stunning him for a few seconds to allow for a couple of more shots.
One strategy is to drop down into the children's play area below, and kick the toy cubes at Adam, using first-person view for increased accuracy. They will knock him over and allow Frank to get a couple of shots in before he retaliates with fire breath. Therefore, run after attacking Adam. Frank can also pick up one of the gumball machines and throw it at him.
When Frank goes through Greg's shortcut, (important in later missions, especially Medicine Run) go to Bachman's Bookporium and pick up the Engineering book along with the Entertainment book. Frank now has practically indestructible chainsaw, the books make it last 27x longer than normal.