If you are new to the series, or wondering whether or not to start watching it, you should start with the OVA's. It is my opinion that the OVA's have the best story, though I haven't seen all of the second Tenchi TV series (aka Tenchi in Tokyo). I believe that the first Tenchi TV series (aka Tenchi Universe) has a little excess filler episodes. But, if you are looking to watch a movie, all three Tenchi movies are excellent, and somethimes hard to tell exactly what they are based apon (ex. the OVA's, Tenchi Universe, or Tenchi in Tokyo.
The storyline of the OVA's is a little different from that of Tenchi Universe. In the OVA's, Tenchi steals his grandfather's keys and sneaks into a cave (that had a locked gate, hence he needed the keys). In the cave Tenchi finds a sword that was supposed to have belonged to his ancestor Yosho, who defeated a demon 700 years earlier, which he trapped in the cave, but it seems to be all rusty. Tenchi accidently cuts through a rock which opens a secret passage. He falls down the passage way, and finds himself in a room with the decaying corpse of the demon. The demon comes alive and grabs Tenchi. He runs like a scared little girl, and gets away.