For software, try Audacity. It's free and is not too difficult to use.
As for getting the audio into your computer, there are a number of methods. The turntables with a digital output, as mentioned in the above post, basically combine three pieces of equipment into one; a turntable, a phono preamp, and an Analog to Digital Converter (ADC). My only gripe with them is that their not the best quality devices in any of those categories. As a matter of fact, I would bet your current turntable is a better source. Also, why buy a second turntable when you already have one??
That said, what you do need is a good audio card. This card, or break-out box, should have line inputs that you can connect the output of either your phono preamp or cassette deck to. As gonk mentioned, your computer probably already has a built-in line input that you can use. For better quality there's a bunch of boxes out there that you can look at. Here's a few:
My personal favorite:
http://www.bhphotovideo.com/c/product/292993-REG/Sound_Devices_USBPRE15_USBPre_Version_1_5_.html It's a touch pricey but built like a tank and adds no coloration to your sound. Add Audacity and you've got a great setup.
Option 2:
http://www.amazon.com/M-Audio-Audiophile...0694079-7791810 A little older but should do the trick. Read the reviews about needing new drivers. Under $100 as well.
There are other options available as well. Look around for analog to digital converters...
Only other thing is that when you get the audio onto your computer, make sure that you are recording it as a .wav file. 16 bit 44.1khz are the standard redbook CD settings.