There is a moment early in the first episodes of "Tenchi Universe" that is the key to understanding what drives "Tenchi Muyo!" in just about all its incarnations. The hero, Tenchi Misaki, and the space pirate Ryoko, are cornered on the edge of a cliff by a battle-mecha. All seems lost. Then the robot slips on a discarded soda can, flips over, and goes plummeting into the ravine below.
This is a very funny scene. In fact, it is even funnier the second time around, because by that time we have learned Ryoko can fly, project energy blasts, throw up force fields and do all kinds of other things that show up her crying out and cowering behind Tenchi to be thoroughly ridiculous.
"Tenchi Muyo!" (meaning "No Need for Tenchi!") in all its forms is about two things: behavior and defeated expectations. It is not about plotting, per se; none of the Tenchi TV series or movies are terribly thick with plot. They are basically about characters with Dickensian attributes -- Ryoko the pirate is greedy and lazy; Aeka the princess, haughty and self-important; Mihoshi the incompetent cop, good-natured and vacuous; and so on. But all of them do care about Tenchi, their somewhat gormless benefactor on earth, and the show is more or less about how their different forms of caring collide.
Seeing any of the other Tenchi series or OAVs is not critical, but it's useful for the sake of contrast. Like "Tenchi in Tokyo," "Tenchi Universe" takes the same basic characters and shuffles the situations around slightly to produce a different storyline. In the original OAV, Ryoko was the "demon" imprisoned in the Masaki family shrine; in this story, it's *Washu* -- but the changes are interesting and usually lead to developments that we might not have expected otherwise.
"Tenchi" is basically slapstick situational humor, but with some slightly somber touches. I always found it interesting that the one remotely sane person in the whole gang is Sasami, Aeka's baby sister, who runs around and keeps everyone else sane while they're all operating at cross-purposes anyway. She is also like Tenchi in that she sees some good in everyone -- even in a pathological liar, criminal, and con artist like Ryoko, which is saying something. In the same way, Tenchi finds it hard to actually tell Ryoko to just get lost -- she may be bad but she's got real emotions under her flirting and bluffing, and it comes out in the oddest ways.
As with other "Tenchi" stories, right from the start Ryoko and Aeka can't stand each other. In "Universe" this is set up through an amusing device where each of them relates an anecdote from childhood -- although both of them tell it completely differently, and it's probably a matter of speculation who's lying. (My money is on Ryoko, but that's only because she's a liar consistently, and not because Aeka is much better.) But the two of them are also shown as being in agreement on something: they care about this young man and will unite to protect him if they have to. Even if they wind up killing each other later. (Much to Tenchi's chagrin.)
I was reminded of the old Shel Silverstein song, about how some kind of help is the kind of help that helping's all about, and some kind of help is the kind of help we can all do without. The "Tenchi" stories understand that very well and spin great humor out of it.