Use A and D or the Arrow Keys to move. Press SPACE, UP, or W to jump. Press X or SHIFT to grab an egg out of the ground. Then press X or SHIFT again to throw it. You only have a limited amount of eggs, but you can get more by collecting candies.
There are all sorts of bonuses to find in the forest. Grab all of the stars to get a bubble shield. You can also get special eggs with bonus powers. Explore each level of Stargrove Scramble to find all the secrets.