Very Computer

I have a system running Fedora Core 2 where every so often the network
dies (ping just hangs, etc.)

Doing /etc/init.d/network restart as root gets it running again.

But something is clearly wrong in the bowels of this thing and I'd like
to learn how to chase it down.

Problem is I don't know where to begin troubleshooting this sort of
thing.  Is there a log file somewhere I can examine for clues?  Is
there some kind of diagnostic network program I can run when it's hung
up that might give me some insights into the problem?

One of the first things to look at is
/var/log/messages
and see if there is any hint of something going astray.

Then make an ifconfig to see if the network interfaces are OK.

Thanks, I'll give those a try -- and I'll probably be back.  :-(