Frogfucius has advised you to visit Star Hill, which is west of Booster Tower.
Explore Star Hill
Go north, then northwest to the westernmost big star with a keyhole in it. Talk to the star flower to the left of the big star, and the big star will open. Go through.
There will be a cutscene. You can optionally talk to the smiling stars that you find, to see some optional dialogue and cutscenes.
In this first area, you need to talk to all five flowers to open the big star door in the west side of the area. First, go northeast and talk to the flower there, then go southwest and talk to the flower south of the big star. Go west from there and talk to the flower in the southwest corner. Go north and talk to the flower there, then go east and talk to the fifth flower. Go southwest from there to reach the big star. Go through.
There are six flowers to talk to in this area. Go southwest and talk to the first flower, then go northwest to the second flower, and then go south to the third. Go east from there to find the fourth flower. Go east to find the fifth in the southeast part of the area, then go north to find the sixth and final flower in the northeast corner. Then go southwest until you can go west to the big star. Go through.
Talk to the flower just west of the big star, then go north to find a second flower. Go northeast and you will find the purple Star Piece. After the cutscene, go southwest until you can go northwest, to find a third flower. Go south to the big star, then southwest from the big star to find a fourth flower, east to find a fifth, then north, and northeast to find the sixth, final flower. Go through the big star to exit the area.
Now that you have the Star Piece, you can go southwest to Seaside Town.