Does your record player have 78 speed? Does it have connections for speakers? Do you have a Hi-Fi system with cassette player/recorder?
The way I do it is to connect my record player to my cassette recorder via the speaker cables and record all my 78s to cassette. Once that is done I use LP Recorder and LP Ripper from click here to record to hard disk and eventually to CD.
Just a thought: Normally a record player cartridge doesn't give enough output for your soundcard, that's why you need to use a pre-amp or headphone output.
But, if it's a really old player, and doesn't have a pre-amp in it, the output from the cartridge might be enough. So connect it straight to the mic-in socket on your PC and see if you get enough signal.
Yes, put them up on eBay by all means but don't forget you may fall foul of the copyright law if you copy them to CD first and then sell the originals. Like yours, my old 78s are in the loft carefully packed in boxes. They go when I go and my kids find them and decide they are not their bag. Mine are all vintage jazz and the copyright has run out anyway, but even so I wouldn't sell them.
I expect someone on the Forum would know of a website that would give you some idea of what their value was (I would like to know as well for my collection). I also have a load of 45s, possibly not so valuable.