Star wars question about the Death Stars?

Here's what I dont get. The galactic empire used the first death star to destroy alderaan in star wars episode 4 a new hope. In the Expanded universe, they destroyed another planet, Despayre, before they destroyed alderaan, the first death star was used on 2 planets.

In episode 5 and episode 6 they didn't destroy any other planet with the death star. In episode 5, the rebels were fighting them, but their death star wasn't built yet. In episode 6, when the galactic empire finally got the second death star, they didn't use it on any planets. They wanted to use it as a weapon/tool to defeat the rebel alliance, but, they didn't destroy any other planets with it....

I mean, the entire film saga revolves around defeating the evil galactic empire, but they barely used the death star! I don't understand it. In ROTJ, they never once used the 2nd death star on another planet, and in the empire strikes back the death star was never mentioned and it didn't really seem to be a part of the storyline...

Were they planning on using it much more on more planets, but the rebel alliance defeated them and stopped them? Or was it just a minor plot point? Wasn't that the whole point of defeating the bad guys? I am very confused because the empire never used the death star after Episode 4, it was barely mentioned in the story saving any planets, yet, the entire story revolved around saving the galaxy from the bad guys. Yet the death star seemed to play a very minor role....

I'm am very confused on this plot point and any help/answers would be greatly appreciated...