This achievement is fairly easy really, if you're a veteran of SA2 or not. On the menu, choose to play 1P and story, then choose the hero story.
There's sixteen levels to complete in the hero side of the story, split between Sonic, Tails and Knuckles. Sonic gets more levels than the other two, but only a couple more.
Sonic levels are, as you'd expect, the fast paced section of the game. However, be aware at all times that there can be sections where you have to do technical aspects as Sonic. During these sections, take your time if you want to be sure of the completion. In particular, the later levels (Crazy Gadget being the main one) has quite long sections where you aren't really racing through like you'd expect. Time, Rank, Score, or anything like that don't matter, so don't worry about going too slow.
Tails' levels are run and gun, or rather walk and gun. The best tip for Tails is to hold the shoot botton (X is easiest) for the good majority of the time, and you'll automatically target anything that flies towards you. These enemies are the biggest pain, not really doing much to damage you greatly, but it can knock you off the level to a certain life loss. These levels are mostly not too hard, as you don't die when automatically hit when you have no rings, and you can regain health by collecting rings. Use hover often (you get it in the second tails level)
Knuckles levels are mainly just irritating, while not being much of a problem to losing lives or not being able to complete. They're more time consuming though. You have to find 3 shards of the master emerald within the level, which can be hidden in various boxes, enemies and underground or within walls. Either that or they can be simply floating around, on a certain path or stationary. They do however, have certain places which they can be, so you can memorise some locations if you die. Get clues by reading the monitor screens, which then all the clues you gather after the first will relate to the same emerald shard. after you gather that, the next clue will relate to another and all following relate to that and so on. This doesn't mean you can't collect the other gems that you don't have the clues to, but they will be either by accident or fantastic eyesight. As you near the emerald, the radar at the bottom of the screen will light up, first green, then yellow, then red, and a chiming sound cn be heard. When you are really close to the emerald, a red bubble with an exclamation mark above knuckles' head to let you know you are very close. use these to close into the emerald. Some Knuckles Levels, especially his final one can take in excess of 10 minutes.
as for the bosses, mostly they are repetitive, with the only real tough fights being the ones between the dark characters once you get closer to the end of the game. In the fight against Rouge, keep hold of a couple of rings and go on the offensive, she can be damaged by punches, or by using drill claw on top of her. For Eggman as Tails, keep on the move and try to stay away from the centre as you dont want to be caught in an explosion. If Eggman goes near the middle, target the green floating container and shoot it to make it explode, hopefully catching him. When he shoots missiles, move quickly to avoid, but the laser needs to you move behind him for a nice cheap shot. Be warned that the laser can damage heavily, and hit you twice if you get caught badly by it. Avoid at all costs. For the final Shadow fight, when you're on the move, use the rings to your advantage. Wait for him to Chaos Control into the middle of a line of rings in the middle, and then use your light dash to damage him. Sometimes it doesn't work, but its the best method that hits more sucessfully than homing attacks.
EDIT: can any negative voters add a comment when you vote and let me know why you voted that way so I can improve it please?