please school me on TIVO, DVR, etc.

I use a DVR that's provided by my cable provider because it's simple and has dual recording channels built into it.

So, I'd recommend starting out that way just for the simplicity alone.  It works just like any other cable box where you plug the cable into it and it has a TV out that feeds your TV.

You then set it up to record every show you've ever wanted to watch and you proceed to have no life for the next 6 months trying to watch all the shows you've recorded.  You will also wonder how you ever lived without one up to this point in your life.

I use the one my cable service has. I can set it to tape shows from an app on my phone, which is great when I either think about something I wanted to watch, or I am out with friends and they mention a show I should watch.

The only downsides are that I no longer know what shows are on when or even what channel - I set it up and forget about it. And sometimes I have too many shows on tape (usually by the middle of the spring). But we almost never watch live TV anymore - even if something is on, I prefer to wait at least 10-15 minutes, so we can zap the commercials. The only exceptions are if we are watching some show with our kids (who are in Maryland and NY) and sharing snarky comments on our instant messaging.

Yes.

A quick note for the O.P. - you need at least a dual receiver (2 lines) to allow viewing of T.V. AND using the DVR when recording a different channel at the same time, since it requires two coax.

DVRing occurs live on the channel that are using for up to 30 minutes, I believe, so you can rewind back to a piece you missed within the last 30 min.

I think ours records what you are watching from the point you started watching until (I'm guessing) it runs out of capacity. Which reminds me - I'd like an intelligent DVR that would say "I know you didn't explicitly say record this but based on the other stuff you do record I am going to grab it for you." It could keep that programming until it needed the space for stuff I explicitly ask it to record or just delete if not watched after a given time. 

Depends on the device.  If it allows you to "pause Live TV" then it is always recording.  But I think you typically still have to hit the Record function if you want the show saved afterward.

Tivo, the original company/DVR box, I think used to offer this feature.  Based on shows you recorded, it tried to extrapolate what you may want to watch.  This was in the era before digital cable...

Then the cable companies realized people were paying someone else, in this case Tivo, for this recording service.  Tivo used to have either a monthly subscription fee, or a lifetime option, for their service.

So the cable companies, as well as satellite, got in on the act, pretty much freezing out Tivo, with the exception of DirecTV (satellite company).  Looking at the Tivo site, looks like they've got agreements with a handdful of cable companies, as well as the streaming services.

Going with your cable company will probably be the easier route, Tivo, if available with your cable/satellite company, may get more features, but you may get caught between customer service departments with technical issues.