Yes you can record your vinyl LP albums and cassette tapes to your computer and then make CDs of them.
There are a couple ways to do this.
The first is to run a cable from the line out on your receiver/amplifier to the sound connections on your PC. You will hook to line out on the receiver and you will connect to line in on the PC. You will probably need to get an adapter plug for the PC end that will have the multipart straight pin plug to go in eth PC and the RCA plugs on end of line from receiver plug in pother part of it. The key will be trying to get volume adjusted on line out and you will lose some quality thru all the connections and conversions. I did this approach until recently and it worked but was not as good as the record had been. You will need a sound program on the PC to record it. Audacity is used often and you could use the sound recorder that comes with Windows. If you wanted to record cassettes it would be same just connect tape deck to the receiver
The second approach is to get a USB cassette desk or USB turntable. Now it is much easier. All you will do is connect the USB cable from the turntable or tape deck to the USB plug on PC. It will automatically set volume going through the USB cable. You will still need a sound recording program, but usually these decks and turntables include Audacity or another program with it. I bought a Westinghouse USB turntable from Ollieís a week ago and it is doing really great recording. Extra part in it was it runs at 78, 45 and 33 where most only do 33 and 45. I have some of my momís 78s and they record good. Doing the first way required getting a program that altered speed of recording. DAK sold me a program several years ago and the ida was good, but not as good as this way.
From a legal perspective as long as you are recording the albums to the computer/CD (you just need to burn the recording to CD using CD burning software with Windows or buying a program or getting another freebie) as long as the copies you produce are for your own use and you are not giving away or selling copies, you should be ok on copyrights.