Funny you say that. As I recall, it was just about this point when I heaved to that I realized I had been dragging a line in the water for awhile. It was embarrassing.
A decade ago, I might have suggested Ubuntu because it seemed like a lot of the open source GIS development was taking place on Ubuntu. It ran fine on RedHat derivatives; it was just a little easier to get running quickly on Ubuntu. However, even then that was probably more true for QGIS and friends than the server side stuff.
Fedora and Ubuntu would both be perfectly adequate. Part of it depends on what makes you most comfortable. If you ever plan to deploy this for someone else, though, it's a good idea to find out what environment works best for them.
If this is primarily (or even partly) about learning, stick with the OS you're familiar with and concentrate on just learning the tools. If your primary interest is using the tools, you might go with the suggestion from @key -- begin with one of the containers. On the other hand, if you really want to know how the pieces go together, there is no substitute for installing those pieces (PostgreSQL, PostGIS, Tomcat, Geoserver) one piece at a time and fitting them together.
(Full disclosure: Xubuntu is my daily driver and Ubuntu server is what we use at work.)
RedHat here in the late 90s, back when you could still find yourself writing a "modeline."
Then Debian in the early 00s when apt was still a major discriminator. Finally, Ubuntu around 2008 just so I was running the same thing I was recommending to family members for ease of use. (At the time, Ubuntu sported the same ease of installation and hardware detection I'd found with Knoppix.)
Now on Xubuntu, but seriously eyeing a return to Debian.
OP here. I was also testing links to MP4 video. Clearly I should have posted this on YouTube instead.
Mod(s), please feel free to dispose of this as you see fit.
OK, good to know, and thank you. I'd forgotten about the OSM tools, but those are definitely tuned for collecting data in the field. I need to revisit those.