Yep, gotta recommend Audacity. I transcribed 1200 LPs last year (from Abba to Zevon). Did most of them at high speed and then downed the speed using Audacity.
The quality of the vinyl varied from ex-pizza tray to pristine.
If I can fault Audacity on anything is that it is too accurate and without some tweaking of the final edits it is too easy to loose the "warmth" of the original recording and end up with tracks that are too crisp.
Noise and pop reduction is as good/bad as any other piece of software. Having said that, I did get Alice Cooper's "School's Out" back into a reasonably listenable condition and it (the LP) has been through about 200 too many parties.
I also did my singles collection (or at least .. the ones that I can bear to listen too now
) These dont have "great" results mainly due to the dynamic range used on them. For example "Good Golly Miss Molly" is almost "unlistenable" as the original dynamic range was produced to enable those toothbrush pickups and six inch speakers to (ahem) blast it out. The transcribed version defies even the most radical of filter settings to get it back to a reasonable balance without clipping and dropouts. More "recent" singles from the 1970's and 1980's don't present problems.
Another problem you might encounter if you are using very old vinyl LPs (1960's and earlier) is that if you are using these newfangled i-something speakers etc, you may have to reboost the bottom end considerably, I mean, we of advanced age had no problems, we all had 2x15" bass enclosures (some of us ..6x etc). So, if you've got early Jefferson Airplane, Dead etc then you are really going to have to do some work to hear them properly on little speakers.
Oh, and finally, you will need to experiment a bit ( a lot ) to get the right capture settings. It's worth the effort. Take a particularly compressed lp and record it flat, at various input levels, until you get the right levels that allows you to edit and expand and etc and prevent clipping.
I found audacity a lot better in the end than the others.