The NES Mega Man titles are famous for allowing you to tackle the first stages in any order you like, and that's the case once more. Though you have complete freedom for the first eight stages, and for various portions beyond, I recommend that you follow the order outlined in this walkthrough for the highest success rate. Some of the monsters pack quite a punch, and you'll want to be ready.
Since Mega Man 2, the blue bomber has acquired a new bag of tricks. The first of these is the ability to slide. If you press and hold 'down' on the controller while jumping, he'll slide instead of jumping. If you have a sticky controller like me, sometimes you'll slide right into pits without even knowing you were pressing the 'down' button. Be careful.
The item system has also changed slightly. You now have a robotic canine as a companion. His name is Rush, and he can change between a springboard, a submarine, and even a jet. As you might expect, the last two of these abilities is unlocked when you defeat various robot masters.
Finally, an added four stages stand between the eight robot masters and Dr. Wily's castle. These are remixes of stages that already are familiar to you, but they're much tougher. You get to fight modified clones of the eight robots from Mega Man 2 now.
That's about all you need to know for the basics, but you may also be interested to hear that this game will let you cheat like crazy.