In all phases except the last one Eggie will fire lasers at you with a nice catchphrase he likes too much. Just run in circles or in a straight line to dodge these. You can spin dash to dodge them if you're standing when he attacks for some reason but that could cause you to randomly fall off so it's not recommended.

Once you've dodged the lasers (and other stuff I'll cover later) enough times Eggman will taunt you. This is the moment to attack him. Use your homing attack but wait till he's fully exposed. You don't need to hurry, take your time.
Hit 2-5 require you to pass more homing attack targets to get to him. Wait till his tail is up all the way and don't spam the homing attack too fast or you'll fall to your doom. Again, don't rush this.

After the first hit he will sometimes charge up a big attack that will explode in a big radius. You should be able to dodge this easily the same way you dodge lasers. If you're cornered it's ok to use a quick spin dash. Only tap the spin dash button tho so you won't go too far.

Probably the first thing people actually need help to dodge. He will do this attack once during each fight. To dodge it stay on the left or right side and quickly switch sides before he would pass you. The lasers can't switch as fast as you do.

Trickiest part of the entire boss fight. Once you get to this point jumping on his moving platforms and riding them to him is the only way to land a hit.
The best way to dodge and jump onto a platform is actually standing still. Wait till he fires his platform and do a nice long jump without moving. Reeeealy hug that jump button. If you do it correctly you should land perfectly in the middle of the platform and can ride it towards Eggman.
To actually hit him I find it best to first jump in his direction, wait a bit (half a second maybe) and then do a homing attack targeting Eggman specifically. He is the weak point so try to aim at him specifically with your stick.
Do this two more times and he'll be dead.
You don't need to worry about him destroying platforms as long as you land your hits. You can miss one hit but after that you can't win the fight and might as well jump in the pit.

Once you drain his lifebar you'll just have to watch out for his last attack. He'll try to crash into you and destroy the platform you're standing on.
Can't really help you with that other than warn you.
Don't spin dash, don't try to hit him, just dodge by walking somewhere else.
This attack is pretty well telegraphed so just watch out for his dipping animation.