You can try a standard turntable connected directly to the PC, but you probably won't be happy with the results. Turntables need RIAA equalization and you usually get that from your amplifier or receiver in an ordinary sound system.
You can get a $35 USB sound card (Behringer UF0 202). It connects to your USB port and has a proper turntable input and can also accept any line level source--tape deck, etc.
Audacity is an excellent way to do the recording once you have the turntable hooked up the right way. It has some built-in processing capabilities--click repair, noise and hiss reduction, etc and they work pretty well. You can spend 15 minutes trying to get rid of clicks and pops in a particular song if you want--one click at a time. It's just a matter of how much time you want to spend on that.
There are software packages out there that attempt to automate the click and pop removal.
I would use mp3 format, but be sure to keep the bit rate fairly high. Probably 192 kbps. Joint Stereo is good. VBR is good. Avoid constant bit rate if you can. Avoid low bit rates. You can set all that up in Audacity. You capture the recording in Audacity and then "export" it as an mp3, with whatever mp3 settings you have chosen.
You could use a USB turntable, but they are not likely to be as high quality as a traditional turntable.