Genesis Points are the premium currency of TerraGenesis. They can be bought with real money, earned by watching ads that come in on transmissions (1 to 3 GP awarded each), earned from live events, earned by reaching habitability, or earned by achieving victory on worlds. There is also a very rare chance that the game will gift a large amount of genesis points (The maximum I have seen is 61)
You can use the game's lagginess to you advantage when trying to get more GP. If the game lags just when you tap the 'transmission' button, tap it a bunch more times. Sometimes it will give you the opportunity to watch a lot of ads for GP that way.
You get 10 GP for reaching 'Plantlife' habitability, 20 GP for achieving 'Habitable' habitability, and 20 GP for achieving victory on a world. This is repeatable, meaning that if you start a new save file on a world you have completed before, you can earn these GP again.
Genesis Points can be spent on skipping wait timers and on buying space credits. Buying 10.000.000 credits for 30GP is the best deal if you are going to buy credits. Since you can keep earning GP on Paradise worlds after you achieve Victory, you can transfer those GP to new worlds and it is very helpful to jump start the latter. You can transfer GP between different difficulties as well.
On normal difficulty, it costs 1 GP per 20 minutes (rounded up) to "hurry" a project, while on expert mode it's 1 GP per 15 minutes (rounded up).
Genesis Points can be bought with real money (but you can easily get all you need by watching ads).
Transfering GP (only relevant for those who have played previous versions of TerraGenesis)
The recent update revamped the GP system, so that you now have a single GP pile that travels with you as you go to new worlds. To combine GP from old worlds, visit those worlds and the GP stored on that world will get added to your GP.
This is how it used to work: All Genesis Points, except for the 20 free ones you get when you start a new world, and the ones you earn from reaching Habitability milestones, can be transferred between worlds. GP you got from watching ads, and from special events (e.g. the 25 GP from Moon landing 50th anniversary) count as 'purchased GP'. When you have both on a world, free GP is used up first. Free GP can be added to the single GP pile too.
To transfer GP, you had to first make sure you are on the planet you want to transfer GP to, not from. Then tap the blue button at the top right of your screen once or twice, until you see the Stats Page button at the top left of your screen turn into a gear-like button that leads to the Control Panel. Go to the Control Panel. Go to Settings. Tap 'Transfer GP.' Select the world you want to transfer your GP from, and transfer it.