I still haven't found what I'm looking for, but I ended up on this page:

https://en.wikipedia.org/wiki/File:Unifont_Full_Map.png

And I downloaded that file, full resolution. It's pretty cool. I open it and blow it up and pore over all the symbols.

The problem is, when I find something I like, finding its Unicode number (half on the left margin, the other half at the top) is very uncomfortable. If only I could click a symbol and get its code... There must be a better way.

Can you recommend a better tool for browsing and selecting Unicode symbols?

Of course, I prefer Linux, but Windows applications will do in a pinch.

TIA

gucharmap

Gucharmap is indeed a good solution. You can view by script or by code and also search. For the real enthusiast, there are the charts to download: http://www.unicode.org/charts/