- The most important improvement concerning fonts is to enable font smoothing - fire one after the other these lines in Terminal:
wget http://files.polosatus.ru/winefontssmoothing_en.sh
bash winefontssmoothing_en.sh
select third option in terminal - with the arrows, then use tab key to select ok and 'enter'

(source here)
- To increase font of menus (source), search for ' ~/.wine/drive_c/windows/win.ini', open it in a text editor, add these two lines at the end, and save:
[Desktop]
MenuFontSize=18
Replace the "18" with whatever font size.
- For a better system appearance install a windows msstyle file in
winecfg (Wine/Configure Wine/Desktop Integration). In this images it's "luna" from XP, which I recommend.

- Increase the screen resolution in
winecfg (Wine/Configure Wine/Graphics), which will make general details (including fonts) larger (if that's what you need: on my laptop appearance details of wine programs are too small by default)

But the images above are taken after enabling font smoothing. That is the most important action, as I was able to test. As I had to reinstall Wine at some point, I also had to restart all procedures here. Using all the other solutions (including installing all fonts as suggested in Shauna's answer - which is very advisable, and to be considered as a part of the solution to this question) -- with the exception of font smoothing, Foobar2000 looked like so:

After font smoothing:

Important: Changing settings in winecfg may reset font smoothing, so it is better to run that in the end, or re-enable it when fonts seem ugly.