Planets will rebel if the other side manages to get four times your influence on them. To stop the process, you can either build a very specific one-use-only improvement (re-education center, I think) that makes the planet immune, or try to build more embassies, Starbases, etc. Remember, planet population and quality also affect influence, and I think that if you only have the first extreme colonization tech your influence gets halved somewhere in the calulating process.

Ahh.  Got it, thanks!

(I'd purchased GalCiv 2: UE last December, but never really got into it for some reason.  I dusted it off the other day and now I can't stop playing it! ).

Keep in mind that starbases can be built by constructors in the cheapest hull in which they will fit and w/o engines.

Once the constructor is done, just adjust which square into which the ship will launch from the planet, launch, and build an influence base, and add influence modules the same way.

Thanks!  I've got a better idea of how they work now.  Since my first post, I built two additional influence bases and maxed them out with all the influence/diplo tech I have at the moment.  It seems to have worked as I created friendly culture bubbles around my bases now. 

Thanks again!