Before We Leave

I would suggest saving this achievement till the end after you have completed most of the game. It most likely will not have naturally popped by the end, but you waste less time having to reload numerous times. I was short by about 20 islands and got the rest using this method.

This achievement can be obtained by reloading a single save multiple times.

A planet normally has between 3 and 4 big islands. You colonize one when landing with a spaceship. The other 2 or 3 with a ship built at a shipwright. You can have more than one colony sailing ship active at a time. So.you can either wait and build 2 or 3 ships or just one.

Build your ship(s) and sail it/them to one of the main uninhabited islands on a planet. Once they are in position, make a save. Have each ship colonize a different island and then reload the game. Rinse and repeat until the achievement pops.

I do not know if the tiny islands count for this. I assume no, but there is no way of knowing as this game doesn't actively track achievement progress. And sorry I'm not starting this game on another profile and reloading a save 100 times to see if they do. :P

EDITED TO ADD: DaCreme197 commented that they colonized 3 tiny islands in a row and the achievement popped. The tiny islands probably do count, so this should cut down on the number of times you need to reload a save if you also park colony ships next to these islands as well.