For my own future reference, I’m writing down the order that I fight the Mega Man bosses. Let’s start with the first game.
The early goal for me is to get the Magnet Beam, which is vital against Big Eye, and the Magnet Beam is hidden in the Elec Man stage behind some blocks. But Elec Man is tough to beat with out the Cut Blade. So traditionally I would fight Cut Man, then Guts Man, then Elec Man. A careful eye will notice that Guts Arm beats Cut Man, so my order is:
Guts Man (with Buster), then Cut Man (with Buster actually), then Elec Man (with Cut), then Ice Man (with Elec), then Fire Man (with Buster), then Bomb Man (with Fire).
I fight Fire Man with the Buster because it’s possible to stand in one place, jumping and firing at Fireman, without ever taking a hit. I fight Cut Man with the Buster because he usually runs right to the place I have to stand to grab one of the blocks with the Guts arm.