At the very start, turn around and enter the Mystery Box for a bonus room. Here, you need to leap from the Mushroom Trampolines up to the Switch Panels, but it’s a lot easier to just use the cursor (L or R) or the touchscreen to activate the panels, allowing you to claim Green Star #1. Move up the steps to a bunch of Switch Panels guarded by Fizzlits. Take your time and switch them all to progress, noting that you can use the trick from the Mystery Box before to turn some of the panels. In the next section, hit the ?-Block on the left side to find a Lucky Bell, then use that to help clear out the Fizzlits in the area before activating the Green Star Ring.
With the enemies defeated, you now have safe passage to collect all the Green Coins and obtain Green Star #2. The next area will have a bunch of Switch Panels, as well as plenty of Fizzlits and two Hammer Bros. It’s much easier to just go after all of the enemies first, then doing all of the panels in peace. Take the clear pipe/cannon to the final area, where you want to break the wooden boxes on the right for a baseball. You should see two blocks extending from the main platform, so carefully jump on those and toss the baseball to the right to collect Green Star #3. Hit the Switch Panels in the area to lower the one ledge, allowing you to get to the top of the Goal Pole.