Here is the full list of all 19 Rebirth of Island achievements.
To knock stones
Have in submission 5 residents
Kill 10 spiders
Set a target for spiders using spider's arrows
Send first signal
Use all tools
Extract rare stone and go on top level
Сomplete the quest of a settler
Fly to orbit on the first try
Kill the first prey
Chop down 16 wood units
Go down to bottom level
Build first house
Survive in first night
Produce 16 items
Extract 8 stones
Going down into orbit
Complete training