If you want to use a cablecard (so that you can tune in non-network stations), you can't go all-Mac, but I've been very happy with my setup:
- SiliconDust HDHomeRun Prime w/cablecard
- Windows 7 PC running Windows Media Center
If you're not using your Mac Mini for anything other than as a media server (likely running iTunes?) I would recommend installing Windows 7 on a separate partition and booting into that. Make sure you use a version of Windows 7 that includes Windows Media Center. You can then run the Windows version of iTunes if you'd like.
In my case I've got a couple of XBox 360's in the house which can interact with WMC and play live TV, recorded shows (all stored on the central Windows machine), and even schedule new recordings.
But if you can live without live TV in the other rooms, you can do the following: On the Windows Media Center 'server', install the Plex server app and/or AirVideo server (they each have pros/cons). Then, get familiar with a Windows script called WTV-MetaRenamer. It will move/rename the TV recording files (which have a .wtv extension) into a TVShowName/SeasonNumber directory structure, as well as append the TV show episode name at the end of the filename. If you're using Plex server, Plex can then store all of the metadata about the show in its central repository. Then comes the fun part...
On your iPhone/iPad, install the Plex and/or AirVideo app. With the Plex app, you'll see all of the information about your TV shows. You can play them on your iPhone/iPad as well as stream them to the Apple TV using AirPlay.
One important caveat: Windows Media Center utilizes DRM on the TV shows that get recorded. This works based on how the cable operator flags them. I have Comcast and most channels are marked as 'copy freely' which is important. I believe that Verizon does the same. But Cox may be different. If they mark the channel as 'copy once' (or something like that), the AirVideo and Plex server apps can't decrypt the DRM, so you won't be able to stream the shows to your iPhone/iPad/AppleTV.
FYI, the Plex and AirVideo server apps do on-the-fly transcoding of .wtv files to MP4 format. If your Mac Mini has an Intel i5 chip, it should have no problem doing this without breaking a sweat. If it's older, I don't know.
If you want to archive your TV shows for a long period of time (and/or don't want to have to pipe shows through your iPhone/iPad to get them to the AppleTV), you could also run an app called MC-TVConverter. It's actually a package of a bunch of freeware apps, with a single GUI. With this, you can schedule it to 'watch' the WMC recorded TV folder and convert (e.g., using Handbrake) your TV shows from a .wtv format to an .mp4/m4v format, retrieve metadata, and add it to your iTunes library. The end result will be smaller files which can then be streamed directly using just the AppleTV (no AirPlay needed). Of course, you can also stream them wirelessly to your iPhone/iPad. I've experimented with this, but do not currently use it.