You probably have no idea what you have to do in the beginning, but that's cool, it will all become clear later (I think). Go down twice for a golden heart, then up right for a health-well, it's well for your health (get it? :)
SKIP THIS PART IF YOU PLAN TO CHEAT
Anyway, also get the dozen or so potions right from where you start, they might help. There's also a chest with a golden heart a screen or two to the right, you can get it by walking even more right and then onto the raised part. Once you get that heart, go back right and head up from here. If you look around a bit, you'll find another golden heart in a chest and a blue potion. If you can't go up any more, go left then until you find a dead man. Search his body to get the light sword. Go all the way back to the beginning now. There are a LOT of other potions on the island, but none are hard to find, so I won't tell you where to get them (although you'll probably need them)
HI THERE CHEATER, WELCOME BACK
Go up three times and left once. A girl here will ask you to kill a dragon, routine stuff for Dink. She'll also give you a potion and a sword, which you don't need, since you have the lightsword. Now go 5x left and 2x up. See the sparkling thingie? Get it. Not that you have any use for it. Also check out the small hole in the upper middle part of the screen for a nut. A secret mega-nut! Time to kill the dragon. Go all the way left to the shore and head down till you can't go any further, then right and just follow the way from there (picking up some potions along the way). Save first though, there's a powerful dragon in the end. Kill it (well duh)
After you've disposed of the dragon go left. The fairy (which looks like a butterfly) will warn you of a powerful enemy. Nothing that scares Dink though. Enter the star and head down. The next part is pretty straight-forward, you'll get some magic, strength and defense from the fairies and there's a small maze, but it's easy to figure out. Keep going and eventually you'll end up with a insanely strong dragon. Kill it (or cheat your way through) and go up.
Continued in part two!