Prepare to record. The problem with vinyl is that it outputs a lower-level signal than other audio components. In vinyl's heyday, a stereo receiver's phono input had a built-in phono preamp that boosted the signal to line level before sending it to the amplifier section of the receiver. Modern audiophiles often buy specialized phono preamps to listen to vinyl in all its glory.
If you want to convert vinyl to CD or MP3 or some other digital file, you have to have a phono preamp in the chain between the turntable and the computer.
Simplify everything. For about $100, several manufacturers make USB turntables (see Amazon.com in Resources). A USB turntable has a phono preamp built in and a USB cable to connect it to your computer. This type of turntable often comes with audio software. If yours doesn't, Audacity is a great freeware audio program (see Resources). Simply connect the turntable to your USB input on your computer, open the program, hit record and start playing the record.
Use your current vinyl setup. If you already have a turntable, don't want to buy another and can listen to your records through your current receiver, you can always connect a cable from the headphone output of your receiver or the line level output (often labeled "preamp out" or "tape out") to the sound card input of your computer. It should preferably be a sound card with standard RCA inputs (see Resources). Some companies make inexpensive external USB soundcards with these inputs on them if you don't have one already installed.
Again, install a program like Audacity, hit record in the program and start playing your record.
Create the tracks. If you are using the headphone output of your receiver instead of the tape out, you need to make sure you don't turn the volume of the headphone output up too loud. Doing so will result in clipping (a harsh distortion in the digital file).
Regardless of which method you use, it is probably simplest to record each song separately. If you don't, you will need to go into the long .wav file of the album side created by your audio recorder, put in CD track markers and then export those tracks to separate audio files. While this sounds complicated, it is actually pretty easy to do, and you can see the flat parts of the .wav file where the markers will go.
For instance, if Audacity is your recording program, use the cursor to move to the point in the audio timeline where you want the marker. Go to the top menu, go to "Project" and then "Add Label at Selection." Repeat this process at every point you want to create a new track. You can play the .wav file by clicking the "Play" button, then hitting pause when you get to the start of the next track. Once you have all the markers in place, go to the menu again and click "File," then "Export Multiple." Audacity now creates new audio files for each track.
Burn the files to CD or encode them to MP3 as you would with standard audio files, and use your normal burning program to put them on a CD or import them into your media player. If you have iTunes, create a playlist with the songs, make sure all the songs have a checkmark next to them; then click "Burn Disk" and "Audio CD."