In order for you to know what to replace, an inventory must be made of what needs to be replaced. Since all radios are different you need to figure out exactly what your radio has inside. you can do this by either looking directly at the capacitors and hopefully the numbers will be still visible. You can also look at the service manual for a schematic for your radio. you can get one here: http://www.nostalgiaair.org/
If they have a schematic for your radio, it will be like mine and list specifically every capacitor and resistor value and any other bit of info about your radio. mine even tells me how to calibrate the AM and FM which would seem impossible without the manual. The manual also show component locations to help you locate the parts.
Next, get a piece of paper to write down the values and voltages you need. I also wrote down conversions from microfarads to picofarads to help when i go to select the new ones. from what ive done research on and what the great people on this website have told me, the paper and wax capacitors are the ones that need to be changed, which is underneath the chassis. As far as the humming noise goes. that is the electrolytic capacitors in the power supply circuit, mine was located in a metal can on the top of the chassis which contains 3 capacitors inside it actually.
Make sure the radio is unplugged before you go poking at things! on most chassis, the metal case is the negative part. but while you're in there, look around and see if there is any burnt out parts or wires that need attention.