Best way I can think of... Assuming the island is NOT in the middle of the ocean, ie off the coast of a continent, you can generally cross this fairly easily, just need to explore a bit.
So best thing, even if its the cartoon island with the single palm tree, an axe and somebody that knows how to turn said tree into a canoe. (Assuming the tree is wide enough, you can hollow it out. Since wood floats very well, you just sail on across.)
Again if its a cartoony based island with no nearby larger landmass (even if its another island), a gun and a bullet only. You'll last 3 days at best, and die from lack of water, might as well make it quick.
And if its a reasonably good sized island in the middle of the ocean - a survival expert (preferably female of course...) and a good blade. I'd rather not have to resort to banging just two rocks together to make stuff, and a good sized knife has VERY many uses...