Upgrade your Mothership, Battleship, and Planetary Defense System!
Upgrading your Mothership (the tall tower structure) lets you upgrade other structures to higher levels. Upgrading your Battleship (the big ship) lets you bring more attackers into combat. Upgrading your Planetary Defense System (the 5 buildings) lets you deploy more defenses to a planet.
Once you destroy a defense it's gone forever so it may take a few attacks to take down a planet.
Once you capture a planet it will turn green and you can collect gold and trophies from it.