If you have had Niche since release, you may have already gotten this achievement due to a bug. But if you do not have the achievement, here's how to get it.
Now, the achievement description says to "Discover all island types" what this means is that you must travel to all 21 kinds of islands in the game (the unique islands from the story mode tutorial don't count). Some players who have done as the achievement says found that they failed to gain it. Why? Because the achievement doesn't track islands in one savefile, it tracks islands in one game session.
So to get the achievement you don't actually have to do it all on one savefile, you just need to travel to all islands in one session! For example, I got the achievement via getting all islands except Overgrown Jungle in Sandbox.
I suggest doing this on sandbox, so just make your nichelings long-lived and winged (give at least one of them heat body for when you travel through cold islands if you aren't going to just use MoveOccupant) and immune to damage and have 1000 food.
Yup, console commands don't invalidate achievements.
I'm unsure if your starting island is counted. it probably is, but just in case pick either Whale Island or Tiny Green.
Here's a list of the islands you need to go to and a terrain description:
Easy Islands:
Tiny Green (small grassland)
Peaceful Meadow (full grassland)
Grass Adventure (grassland-savanna-swamp mix, may have a very small amount of jungle)
Archipelago (grassland island chain, seems to only be accessible via a Peaceful Meadow northern port or Whale Island)
Medium Islands:
Burning Savanna (full savanna, no accessible water)
Rainforest (jungle-grassland)
First Snow (snow-grassland)
Overgrown Jungle (the "thorns" island in Story Mode, majority jungle with patches of swamp, very small amount of grassland, no accessible water)
Swampy Hill (swamp, small amount of grassland)
Grass Mingle (grassland-savanna-swamp mix, possible small amount of jungle, can rarely spawn jungle and ape trees
Whale Island (whale.)
Hard Islands:
Oasis (full savanna, accessible water)
Crossing (small grassland, north port always leads to Home Island)
Summer Mountains (grassland-snow mix)
Long Winter (the "frozen" island in Story Mode, full snow, one-tile line of accessible water)
Sleepy Reeds (full swamp island)
Jungle Gate (southern side is grassland, northern side is jungle, small patches of swampland)
Killer Islands:
Deep Jungle (full jungle island)
Home Island (red trees, yellowish grass, bunch of nichelings)
Frost Lands (full snow island)
Deadly Hills (full grasslands)
I really highly suggest writing down what islands you have been to. Whale Island is MVP always go to it if the port is available. If you have been to every island but the achievement hasn't appeared yet, try to wait a few days or go to islands that could be mistaken for each other such as Rainforest-Overgrown Jungle-Jungle Gate or Long Winter-Frost Lands.
Hope this guide helps you!