Get into it and learn a little bit as you go each time. Practice makes perfect. That's pretty much how you get through everything in this game.
More specificially, on the first form, just on that third egg and start pounding away at the fire button like your life depended on it. You can get about 15 of the roughly 25 hits needed to take out his first form that way, and if you get good at it, you can take out the whole first form before you even need to leave that egg (by jumping up and timing your shots to hit him on the downswing).
The second form is easy if you remember to keep track of several things: you want to keep track of roughly how frequently he shoots those eye-beams. If he hasn't shot one in awhile, try to avoid risky jumps if possible so you don't land right into one of them. Same goes for those shy guys or whatever the hell they are that pop up. Finally, try to make sure that you are at the very far right of the egg platform before you make your jump if you have to make a really far jump. I can't tell you how many jumps I missed not paying attention to that, and it's easy to lose track in all the confusion.
On the third form, the biggest problem becomes the fact that you can no longer touch the front of the eggs. If at all possible, you want to get your final hits on the 2nd form when you're higher up on the screen, so you can give yourself some time to adjust by having a relatively easy first jump. This requires you to count the number of hits you put on the second form, however. I guess I'm just naturally good at that kind of thing.
Keep in mind also that The Kid is very responsive in the air and you don't need to do a double jump right away. Get creative with your jumps and you should be able to avoid the missiles, and everything else is basically the same. For this form, I would go with the rest of the crowd here and say just keep firing shots to the right when you get a chance but otherwise focus on avoiding things.
Hope this helps a little.
