Lots of good suggestions here. Here's what I've done and what I have been doing:
First shot at digitizing my vinyl was using my Rega RP1 along with the Rega MiniPhono Amp/USB ADC. This was a very simply setup: you connect your turntable to your amp as normal and instead of hooking up the amp's analog outs, you simply hook up the USB cable from the amp to your computer. Adjust the levels on the amp, and its basically a dedicated ADC/Mic In for your computer. I would then use Audacity to record, edit, label, tag, split and convert my files to FLACs. Only problem with this is that the MiniFono's ADC can only output 48/16. A lot of people will argue that this is enough to rip an LP faithfully. I don't disagree and of course 48/16 is more than enough if you are looking to put this stuff into CD form, or into a FLAC/mp3 that most DAPs can handle.
However, the quality of the ADC is also in question. I do think my original rips sounded fine, but still lacked alot the warmth and detail of playing the LPs on the same Rega, through the same MiniFono (but straight analog out, no A-D conversion), through my home stereo system. So what I decided to do was get myself a sound card with a GREAT ADC and with Analog In. So I got myself the E-Mu 1212m PCI. This card is a few years old, so you can pick one up for fairly cheap. The ADC on the card is fantastic, and it does have stereo analog in's (although they are 1/4" Ins and not standard RCA ins, so you'll need a goof 1/4" to RCA stereo cable); it also has the ability to output at 192K/24-bit. Overkill for vinyl? Sure, but I'll explain why this might be a good capture setting. So the process of hooking everything up is similar: you hook up your TT to your amp (any amp) and hook up your amp to the 1/4" analog ins of the EMU via your speacial cable.
The recording process is a lil trickier to set up at first (hell, setting up the card to output AND input sound is tricky itself at first), but once you set up the recording process once, its easy to do from there on (you essentially have to set up appropriate "profiles" on both the sound card driver and recording software, but once you save these, you are set - if you go this route, you can always ask me for my profiles which are set up to rip LPs and playback music at the highest bitrate). I am still mastering the card, but so far I've had to keep the card set to ASIO, which means Audacity wont play nice with it. This is fine. The card comes with Sonar LE which can record through ASIO and at 192K/24-bit (again, I have profiles setup to do this, which I'd happily share). Once you record an entire side of an LP, you then output the file as a WAV, and edit it in Audacity. Audacity has descent algorithms to help clean up recordings from clicks and pops, but before diving into editing in Audacity, I prefer using DeClick and DeNoise software (that their name) to help clean up the WAV.
Ok so back to the topic of ADCs and recording at 192/24. The most important thing, whether you buy an EMu or not, is that you buy a card with a great ADC (it doesnt have to be 192/24 capable, 96/24 would be fine) - this is the KEY to getting the most out of your vinyl rips. Even if you want to burn 44.1/16 CDs or do 320k mp3, both will sound BETTER if the source they are being created from is of better quality. So having a really high quality ADC is most important. Recording at 192/24 (which I do, just cuz I can and I have the HDD space), or even 96/24 is just a good idea. Sure its much more than is needed to completely capture the fidelity of an LP, but it also gives you wiggle room to "assure" yourself you care capturing EVERYTHING. Most good cards with great ADCs have ADCs that can capture at 96/24 and some like the EMU can do 192/24 very very well (although the converse is not true: just because a card can capture at 192/24 or 96/24, does not mean its a great ADC, in fact you COULD argue that for most consumer grade cards, a card with an ADC that can top out at 96/24 is more likely to do 96/24 extremely well, than an average card that does 192/24 - for your purposes you would prefer a card that does 96/24 extremely well, than a card that can so 192/24 just OK; thankfully, the EMU does 192/24 extremely well and its cheap because its a bit old).
Once you have a 192/24 or 96/24 FLAC, you can just get something like JRiver to burn the FLAC to CD and/or mp3 while doing the downsampling automatically. The point here is the better the rip, the more accurate your CD or mp3 will be. This also applies to the idea of getting a new TT with a built in USB connection, in general, not only will the TT/cartridge usually be a lil lower in quality, the built in amp and ADC will also be a lil lower quality, making the rips sound far inferior than what you are used to listening to on your regular TT/system. Also, by recording at a higher sample/bit rate, you can always listen to the full fidelity of your LPs when you play the FLACs on you computer, so you assure yourself that you get the best, most accurate representation of your LPs possible for a given format (FLACs on your desktop, CDs in your car and mp3 on your DAP). So if you have a good TT and a good amp already, just concentrate on getting a good card/usb stick/external DAC with a very good ADC.
Again, I recommend the E-Mu 1212m PCI or one of its bethren! Very economical and a super ADC for the price! (Of course if you got some $$$, you can get something much better, but I find that my 192/24 FLACs sound as good as the original vinyl).
So wh