Ok, seems like there's quite a few people with these problems. I'm not a real techie so can't see any common thread with what's going wrong, but a few other people have posted solutions to potential problems.
For starters, Darstaed came up with the fix that let me play Entrenchment and Diplomacy:
https://forums.sinsofasolarempire.com/419008
His post is about halfway down the page. Commardenoah, if I can follow it, it has to be pretty basic!
The new patch has sorted out the resolution problem for me and HD works.
However, if you're still stuck if you posted what graphics card you've got, sure someone here would be able to tell you how to set it to work (if it is the same problem).
Mordith, sorry you're stuck. If you've tried starting in different resolutions/windowed mode and the ATI (is it control centre? Been a few years since I had one!) settings I'd be just as lost. As shots in the dark, from what people have put here mainly you could try
* Delete AppData\Local\Ironclad Games\Sins of a Solar Empire\Setting\rebellion.user.setting
* Verify integrity of game data cache in Steam (forces download of broken files)
*Delete/disable your codecs (can't find the post now but I'm SURE someone said there was a compatability problem that screwed up their system.
Failing that, go out and kill a prostitute. It won't make Rebellion run, but should vent some stress.