How can I find an electrical problem without hiring someone?

look for a loose wire,

buy a multimeter,

one that can check for Resistance, [for finding shorts ] ac voltage [for current flow]

,dc current, [ used on cars]

a buzz in something normally indicates a loose connection, on a wire, making the voltage jump to the the connection, this is called a cornea, [high peak] making a buzz in things

This sounds like an interference of some kind. It is possible you have an appliance running that interferes with the other appliances. Fans are noted for this. However, this interference can be caused by a neighbor also; running something that gives off a short wave. This problem can also be a result of an appliance not grounded correctly. The appliances that are making this noise, try and plug them into another wall socket and see if this helps. If all fails, call an electrician and maybe the electrician can help.

Check all the connections in your electrical system, especially grounds. Many times, electricians used push-in terminals on the backs of switches and outlets, and this makes a very poor connection. I've seen many of them fail. If all your outlets and switches are old and original to your house, now may be a good time to replace all of them with new ones, using the SCREW terminals -- NOT the push-in terminals. Also, check all wirenut connections in every box, making sure they are tight. You may even want to run a completely new circuit directly to your entertainment center from the breaker panel, to assure the best connection possible, without the electricity having to pass through many outlets and connections along the way.

Try to disconnect the transformer to the door bell. A transformer may very well cause that.

try flipping the circuit breaker on and off a couple of times.