The biggest frustration for you will no doubt be understanding the computer angle. The actual boxes we use to tape now are very simple; they work more or less just like a D8. You just need to use the right settings (WAV file, 24bit/48kHz, etc.) and you're good to go. They're lighter and easier to power, too.
I got back into taping in 2009 after not taping since 2000, so I missed a lot of the transition from physical media to cards. The computer stuff at first is daunting. Where it used to be that the most important thing was to just get a tape made and set the levels right at the time, now all of your time is mostly on the back end. Your levels at the show, within reason, matter very little, as long as you peak around -12dB or higher.
So basically, your front-end is the same. Set up stand, connect mics to preamp, preamp to deck, hit record. (Oh, or you can get a deck now that has decent internal preamps... though I guess the DA-P1 did too...)
AFTER that, it's:
1. Move WAV files from SD card in deck to computer.
2. Edit. For this you need a program like Audacity or other editors. Some guys do little more than boosting the recording's levels to peak at -0.1dB, setting the recording to fade in/out and cutting the thing into tracks. But with the computer, you can pretty much go nuts on EQ, decoding to M/S, adding reverb, adding various tube/tape effects, reducing peaks, reducing noise between songs, compression, etc. etc. etc. My suggestion would be to at least start by just cutting tracks and amplifying to avoid massive annoyance.
3. Export. Basically a part of "Edit", this means exporting the edited WAV file into individual tracks in the FLAC format. Most people also like to still downsample to CD format (16bit/44.1kHz) even though they record at 24/48 or 24/96. Software like Audacity can handle all of this in one step. It's not hard to do - just clicking a setting or two - but it takes time.
Some people (especially Windows users) use multiple programs for this stuff - Audacity or something else for editing, CDWave to cut into tracks, and then Trader's Little Helper to do the downsample and FLAC and to make a "checksum" file which is how people now verify that a digital recording is unaltered from the original.
4. Listen. FLAC can be listened to on a variety of computer players such as Foobar for Windows, WinAmp for Windows or Cog for Mac. They can also be burned to CD, of course. For listening on the go, you either need a portable that plays FLACs, or use a program to change the FLACs into MP3 (lossy, but sound pretty darn good these days at the higher bit rates) or, for Apple iPhone/iPod users, ALAC (lossless Apple compression scheme)
5. Upload. If you want to share the recordings the main way to do it is a BitTorrent site like bt.etree.org (for bands that allow taping) or DIME or Trader's Den (for bands that maybe don't), or uploading to the Live Music Archive. I'll call all of that "Lesson 2" and let someone else take over on that end of things