I've been doing the same thing for myself for awhile now.
The tapes are the easist, you probabally have cassette palyer already.
For the vinyl LP's, of course you need a Record Player.
For software, download Audacity (free at http://audacity.sourceforge.net/)
You'll use this for both.
1) cassette, you'll want to connect your player output or headphone to the audio-in on your PC.
2) You might have to enable or turn up that input from the control-panel, sound controls
3) In Audacity, choose the correct input from the drop-down list.
After adjusting volume levels of your output and input levels you'll simply click the record-button to start recording. (adjust inputs to just below the red-line mark for best volumn/clarity)
You'll have to decide whether you want to record each song seperately (more work at the computer), or record the whole side as 1 and then mark where each song begins (a bit less work IMO). You'll have to follow the audacity directions to download the "lame.dll" (the mp3 encoder plug-in). Then you'll export the project, and you'll get .mp3's.
For a Record, the only issue you might run into is A) You need an AMP to connect the record player to, B) you can't just plug it into the PC like the tape player. Use the Amp's output (or headphone if necessary)
I got a record-player that has a USB output connection that plugs right into the PC usb jack. That makes it a little easier to connect, but otherwise it's the same process.
Once you get used to the Audacity software, the process goes pretty quickly after the 1st couple of tapes/LP's.
I hope that helps explain what you'll need to do.
Good Luck.