Viewing/recording/encoding digital tv (DVB/ATSC) is possible on a pc with a minimal amount of investment.
You'll need a TV Card or TV Dongle plus a standard aerial (rabbit ears). Check for linux compatibility. Expect to pay about $50.
On linux, Kaffeine will allow you to download local tuning frequencies, tune in your channels, watch tv, view the programming schedule (EPG) and schedule recordings. Although it is part of KDE, you don't need to be running KDE to use it.
Me-TV is an option for GTK users.
On windows, your tv card/dongle will ship with a proprietary application to view and record broadcasts.
Recorded tv broadcasts will be stored as MPEG Transport Stream .m2t files. These can be played with mplayer, vlc and many other video players. These files contain uncompressed audio and video data and can become quite large, so for any broadcast you wish to keep, encoding these files is a good idea.
Depending on what your local tv stations broadcast, your MPEG Transport stream files can be interlaced, have differing dimensions to their aspect ratio (squashed horizontally) and most importantly contain missing frames.
Think of a tv broadcast like a livestream or data from a multiplayer game server. UDP rather than TCP. Any data which is not correctly received by you will not be rebroadcast. "The show must go on!" You may have seen this when a digital broadcast breaks up into small squares during a thunderstorm. Your recorded transport stream file will be missing these sections, and if there's enough of them, they build up over time and the audio will desync.
To fix this, use the cross-platform Project-X tool:
$ java -jar ProjectX.jar /your/.m2t/file
This will correct missing sections and ensure the audio stays in sync.
The output from running Project-X will be an .m2v video file and an .mp2 audio file (or similar, depending on what your local stations broadcast). These can then be imported into your video editing software for encoding.
If there are advertisements that you wish to cut from the broadcast, depending on your video editor you may need to recombine your .m2v and .mp2 files into a single .mpg file, then import the .mpg before editing. This is true for AviDemux (cutting out video doesn't cut the corresponding audio otherwise).
For tips on encoding your video, see Video Editing.