Navigating the first screen without unnecessarily slowing down can be tricky, especially because you want as many eggs as possible for 1-8. Other than that, Fuzzies are randomly generated, so you either need to clear the path in front of you by shooting an egg or two, by spitting one of the rolling enemies into the Fuzzies, or by just praying for good spawns. If you do get Fuzzied, jumping as soon as possible after touching the ground decreases the slowdown.
Be prepared for this if you decide to spit a Melon Bug (rolling enemy) into the Piranha Plant. Occasionally, the Melon Bug will clip right through the Piranha Plant without actually damaging it.
At the end of the fuzzy section, there is a pesky 1-up Shy-Guy who flies down right in your way. He can be cleared with a well-timed up-tongue.
You want to eat as many enemies as possible in the beginning, though this can be tough due to the odd slopes and nature of the rolling enemies. To shoot the first flower, simply push egg without jumping on the log.
There are two strategies for dealing with the flashing red eggs. One is to shoot them at predefined targets, such as one of the two middle flowers and the piranha in the fuzzy section. The other is to switch past those eggs and simply carry them into the goal with you.
The sky shots for the final 4 red coins are probably the most difficult part of this level (besides fuzzy trolling). The coins make a zigzag /\/\/\/\ pattern, with the red coins at the top point of each carrot. To shoot them, simply hold up until the screen starts scrolling, and then jump and flutter while firing. It is crucial to listen for the sound of four red coins being collected to know that you got them all. It is possible to shoot all four in one set of flutters, but it is substantially easier to get them two at a time, as shown below.