First off, it depends on your audio hardware. Most turntables do not have the proper output voltage level (RIAA equalization) that is required for the recording without an external signal amplifier. Absolutely no turntable of any worth with respect to sound reproduction will have this ability. In this case, you will need either a mixer or a preamplifier with the appropriate input capability (either moving magnet or moving coil, depending on your turntable) and the ability to "loop" or export the signal being conducted (in this case, your songs on vinyl) to an external source (your G4's audio input).
Assuming you have a turntable that can accurately reproduce the source material (your records), and that you have the proper hardware to increase the signal voltage to near RIAA standards, as well as export the signal to your computers audio input (via a mixer or a preamplifier), then you need an audio recording program such as Audacity which will store the audio onto your hard drive. You can either store the audio files as mp3's initially, or record them as .wav (or whatever else your software allows) and covert them to mp3 in the interest of saving space. For that, I recommend Apple's "lossless" importation feature which is integrated into iTunes.
If I were you, I'd be more concerned with the quality of the turntable, cartridge / stylus, tonearm geometry and alignment, and preamplifier abilities than I would with the actual "recording" process.