Time until Mega Man 5 release!
This particular Mega Man game is one of the most challenging with very interactive sprite powers. Three items are of absolute importance: The Air Dash sprite sheet, the N (Chimera) armor, and the Dual Buster. The last two are needed to get items, but the Air Dash is essential to even reach some of the bosses, making it the first item you should ever attempt to get. I'm am not an avid advocator of entering a board to get an item, and dying to exit the stage, but some beginners may want to consider doing this to get the Buster and N sprites. I mention this because the path I suggest following puts off getting these items until later into the game, which in turn forces you to get other items even later.
Mega Man Sprite order is important, as it always is. It wouldn't be a Mega Man game without and boss order to follow. However, in MMX3, the correct weapon doesn't do tons of damage like you would expect. What's important in MMX3 is that the correct weapon cancels the bosses' attacks. Some of the bosses attack you in such a way that they nail you with a barrage and don't allow you to retaliate. That is why following the order is a lot wiser to do than before. Only certain weapons will give you the opportunity to press your attack on the sprite.
Therefore, the best Mega Man sprite sheet to rip first is Blizzard Buffalo. Not only is he the (possibly only) easiest Mega Man sprite to finish off with your normal X-Buster, but housed in his stage is the essential Air Dash power up capsule. I will proceed to each stage here. I will list how to reach all the items within each stage, as well as how to beat that particular boss. The listing of each stage will be in the proper sequence one should take to get through them. Listed in brackets will be what is needed to get that item. A weapon name with a star next to it means the powered up version of that weapon (requiring the Dual Blaster).