I'd start by looking at Audacity which can record an analogue signal from from a tape cassette deck or turntable via the line in input of your computer's sound card or sound circuitry.
Once you've captured it the file can be saved as a WAV file which can be burned to a CD, which will replay on a domestic CD player, or to an MP3 file which can be imported into iTunes.
In addition to a regular turntable you will need a preamplifier (for example the phono the input of a hifi amplifier directed to the tape outputs of the amplifier and thence to the computer. There are USB connectable turntables but these seem to be of middling quality. You'll probably get a much better sound from an old hifi turntable.
Audacity can record each song on an LP as a separate file manually -- I wouldn't trust programs which claim to distinguish between individual tracks and the silence between them in order to split an album side into song files.
There are programs more sophisticated than Audacity which will allow you to retrospectively insert track breaks into a whole side of an LP.