I have a large collection also. Just was thinking I should sell them at the garage sale this month but cannot seem to part with them. Some were my grandmother's. Really need to research on how to put them on a CD.
The process is really quite simple, just time consuming. You need an audio capture software program like Audacity (free to download online). You also need a standard male to male audio cord -- depending on your record player's output/headphone jack usually a 3.5 mm on both ends is what you'll need. Connect the cord to your stereo out and your computer mike and run the software (you need to configure a few initial settings the first time to set the file type and output recoding quality and a few other things). Then it's as simple as hitting record on the software and manually starting the record player.
You just have to sit through it to make sure the record doesn't skip - otherwise your end recording will also record the skip over and over... If that happens (using Audacity anyway), there is a way to edit that part out of the end file manually. Once the record is finished, you have to go through a few steps to finalize and save the audio file on your computer.
You can burn those files to a blank CD, or save them to an mp3 player or whatever is your preference. The process may take a bit of trial and error at first to get comfortable with it, but it gets easier with practice.