Personal computers are highly modular by design. The most powerful trouble-shooting technique is to isolate the problem to a specific component by trial-and-error. Swap compatible components and see if the system still works. Try different peripherals on different machines and see if the same problem occurs. Make one change at a time.
(2) "It's the cable, s-----."
More than 70% of all computer problems are related to cabling and connections. Ensure all cables are connected firmly. IDE and floppy ribbon cables and power cables can often go loose. Ensure microprocessor, memory modules, and adapters such as video card and sound card are inserted correctly and didn't "pop-up" during transportation.
(3) Don't be frustrated!
Don't be afraid of computer problems. It is often the best opportunity to learn. Trouble-shooting is part of the fun of owning a computer. Imagine the satisfaction you could get by solving a problem yourself.
Of course the fun could ran out quickly once you are frustrated and have spent too much time on the same problem. If you feel frustrated, it's time to leave it for a while and go back with some new ideas or call someone who can help. Rule of thumb: You shouldn't spend more than three hours on the same problem at one time.
(4) Take notes!
Take notes of what you have done and all the error messages. You may need to use them later. For instance, when you see an unusual blue screen with an error message, copy the entire message onto a piece of paper. In many situations, that message may point to the right direction in getting the problem solved quickly.
(5) Take a look?
It's OK to open a computer case and take a look inside. There is only 5V and 12V DC voltage supplied to the components outside the power supply. Those who have never seen the inside of a computer are often amazed by how simple it looks. Of course, still always power down and unplug the power cord first.