Health of ships is obtained by toggling the sensors screen on and then toggling the info button (an 'i' button at the bottom of the screen.) This will give you a green/yellow/red cross to display approximate health.
Health of the planet is obtained by looking at the top left corner of the screen, there should be a box with several wiggly bars in it. The longer the bars, the closer the current population is to max population. You need to eliminate all population to clear the planet.