Ah! I just went and tested it on my phone to be sure, but it looks like downloading and installing the new version automatically keeps both your saves and your persistent data (like collected colors and stuff)!
(As a small warning, sometimes things in the game may change between versions in a way that breaks the save. I recommend trying to avoid splitting a playthrough over multiple versions if you can avoid it! Nothing wrong with trying though, and it should be fine between most small updates)