Ping performs the most fundamental tests of network operation. TraceRoute can help diagnose routing problems. TCPdump was designed to study TCP operation, but can also be used as a generic packet analyzer. Dig is the most useful tool for studying DNS.
The real trick of successful network troubleshooters is this - they understand the protocols and design well enough to know what should be happening in any given instance. That doesn't mean memorizing all the RFCs, but you need to know where to go to get information when you need it. Once you know what should be happening, break it down into simpler steps or components. Now apply your American ingenuity and devise some Goldburgian setup to test each little step. Narrow down the problem and test it again. Keep a notebook. Write down everything you've tried and what the results have been. Keep this up and you'll learn a lot of stuff you thought you already knew. Eventually you'll either fix the problem or find a good reason to blame it on someone else.