A lot of the German TV shows and movies (especially the ones produced by public TV channels) will actually have German subtitles produced for them to aid the deaf ("Unterstützung/Untertitel für Hörgeschädigte" or Keyword "barrierefrei").
All German DVDs (movies and shows) that I have bought so far have subtitles (in German).
YouTube let's you enable subtitles, and most .MKV files will have multiple subtitle layers which usually include German for above mentioned reasons, sometimes you can even download subtitle (information) separately to the video files.
VLC-Player can let you switch between subtitle layers which also helps with a quick translation by switching to English if you didn't catch the meaning at all.
Here seems to be a nice list of publicly available shows that have subtitles (of ARD):
There is a "UT" button (lower right on the video) that you can use to turn on the subtitles in the Das Erste Mediathek (see example). But these might not be available outside of Germany :( .