I poke at the problem until it goes away, using my vast knowledge of Strange Shit I've seen in this business since I started.
If the problem doesn't go away, or I'm still in the dark at the end of my poking I ask a young person. Or Google.
The above steps have always solved the problem - If they don't I'll post a question here in the hopes that someone else has seen this new-to-me variety of Strange Shit before and has the answer written on the back of an old cocktail napkin somewhere in their bottom desk drawer.
While the above sounds humorous it's honestly the process I follow. My poking is more scientific than blindly running commands (usually), and always begins with what Rob Moir suggested:
Define the problem fully.
Often this points you at the solution, but if not it at least gives you an idea of where you should look.
Armed with the definition of the problem I ask myself the Key Question of System Administration:
What Changed?, and if I can identify an intentional change I put whatever changed back the way it was to see if the problem goes away.
After that it's down to the scientific method like Iain mentioned, methodically working through and investigating the issue, isolating probable causes, identifying possible solutions, and testing them.
Ideally you would do all your testing in a dev environment (and sometimes I've gotten lucky), but often what broke is production and the right answer is the "quick fix" -- Reboot the production box and make it work again. This is always followed up by making the problem happen in the development environment and learning Why it broke, and trying to make sure it won't happen again.
Asking a young person is surprisingly not on anyone else's list, nor is the more general "Bother your colleagues".
In my experience when all the senior admins are stumped sometimes explaining the problem to a junior admin (or someone outside the tech team) is helpful - It makes you think about the problem, and sometimes they have ideas that you haven't thought of.
Server Fault is also a great resource for learning. I've seen solutions here for problems I haven't yet had, but can anticipate happening soon. When I do have those problems I know the answer already, because I read it somewhere on the internet.