Solving problems as they come up is probably as good a way to learn as any. There is lots of material online for free on the subject. One of the most crucial skills for an admin to have is the ability to use Google to locate answers (sometimes it's easy, sometimes it requires cleverness and much iterative searching).

We've just had a problem with sites loading slowly/not at all and I had no idea where to start looking for the problem.

And you can google for "problem solving" to get some ideas on where to start. The general strategy of narrowing down the problem is almost always applicable at the beginning.

For example, if I had a site loading slowly, I would want to rule in or out network problems. If I can log in directly to the server, use lynx to browse the site directly with no slowdown even though it's slow to external clients, then I've probably ruled out a great many things that can go wrong with server configuration. OTOH, if a local lynx browser sees the same slowdown, then I've probably ruled out a great many things that can go wrong between the client and the server's NIC card.