Hey,
I'm also a huge fan of these kinds of games, and have a couple recommendations. Civilization itself isn't very accessible, but there are a few games which are similar.
If you like science fiction, check out
$1. This is basically a huge space sandbox, which is quite complicated but happens to be almost completely accessible. It's not really for the faint of heart but there are plenty of people on /r/aurora who are willing to help you out, in addition to the game forum I linked.
Again sci-fi, look into the
$1. This is a text-based port of Master of Orion, which was basically the sci-fi answer to Civilization back in the day. It's not dumbed down in any way, and combat is a little hard with a screen reader, but the whole game is there if you can tolerate the command line. The link is to my Dropbox, with a ready-to-run version of the game, including a manual and command reference.
Finally, look into
$1. This is a much more casual game than either of the above, but is at least themed around the same kind of worker/building/production cycle as Civilization. It's fun if you want something you can enjoy in your web browser, though the details are of course quite different from its obvious inspiration.
I hope that one or two of these at least catch your interest. I"m happy to help out and provide guidance if you have any questions.