Is there anything I should know beforehand? WINE is giving a bit of trouble, as it usually does, but all the fonts are FUBAR.
WINE shouldn't be necessary for the editor (although of course it will for Civ3 itself). The editor ought to run without any issues (or at worst, any serious ones) with Java on Linux.
For Civ3, I seem to recall being able to install Vanilla but having some issues with Conquests, as well as some general issues due to the copy protection on the CD. However, I never really pursued it that much since I was only playing around with it and still had Windows as a main OS. Others will be better able to help with getting Civ itself running. And it may have improved and/or changed since I tried it.
As for the fonts, the editor supports three fonts that are present on
most Windows, Mac, and Linux platforms. Tahoma is the default, with Trebuchet MS and Arial being other fallbacks. If the default isn't working well, you can change it in Settings, or if it's
really not working well, you can open the civ3editor.ini file which will appear in the same folder of the editor after it's run for the first time, and modify the fontChoice line. I'd recommend setting it to fontChoice=Arial, as I believe that's the most common of the three on Linux (although on systems with Tahoma and Trebuchet MS, they tend to look a bit better, and IIRC they might be more common on Macs).
I'd also be curious what it looks like and which distribution of Linux you're using if the fonts are really off. It looked okay on my main Linux VM, but that one is from 2009 and has KDE 3.5, so it's pretty out of date by now. I'll probably create a new, much more up-to-date one once the new Linux Mint comes out later this month, and do some testing on that.
Technically there is a fourth font choice in the editor, SimSun, but it's only there because Chinese BIQs needed a font with Chinese character support. If it's present on your system, it's a valid option for non-Chinese BIQs as well, but I don't expect it to be as common as the other three.