Lost Universe: Volume 1's purpose is mainly to introduce the characters, universe and concepts - and it seems to do a fairly good job at that. The first episode, "A Blade of Light Shines" starts off with Millie, a Jill-of-all-Trades who, while investigating a black market auction, ends up having Kain Blueriver and the A.I. for his ship Canal drop in on her. While the three characters work together, Millie learns that Kain is a special operative for the Universal Guardians and wields a psi blade (a weapon that requires extreme mental strength).
"The Goddess Flies," the second episode, teaches us a little about Lost Ships and we learn that Kain's vessel, The Sword Breaker, is one of the fabled Lost Ships and that it was given to him by his grandmother. Millie shows back up claiming to be the "best cook in the universe" this time, instead of her previous claim of being the best detective. It seems her goal is to be the best at something - nothing in particular, just something. The focus of this episode has The Universal Guardians hire Kain to hunt down a ship that has been in development for years. This ship is supposed to be on par with the technology found on Lost Ships, but unfortunately it has gone rogue.
The third episode, "The Kitchen Dances," puts the crew of the Sword Breaker on an escort mission that has them taking a witness to the other side of the galaxy before the imminent trial. The only problem is that the witness is going to testify against a major crime lord - so, of course, it isn't going to be as easy as just jumping across the galaxy because first they have to get their cargo to the ship.
After that, the crew goes to the beach of San Zania while Millie attempts to win her 100th pageant. In this episode, Millie tries her hand at activating and controlling Kain's psi blade, but finds it to be much harder than it looks. But, like most things in this series, there seems to be a bit more than just a contest going on - for some reason, an assassin seems to be targeting Kain.
The fifth episode is "Flames Roar" and the team is helping out the security for a man-made planet. Unfortunately, it seems like the evil organization, Nitemare, has taken a special interest in Kain and the Sword Breaker. Now Millie and Kain have to escape an underground passageway while the Nitemare boss tests them and Canary looks for more weapons for the ship.
Lost Universe: Volume 1 ends with Kain and Millie having to infiltrate Highball, a female-only nursing school. So, of course, Kain has to dress up as a woman. Their mission is to protect two student groups who are fighting with each other. When a fight breaks out between the groups, it is up to Kain to not only stop the conflict but also inspire a whole graduating class.
Like I said, the main point of Volume 1 is to set up the characters and style of the series. The more comedic feel of Lost Universe makes it a good, fun anime to watch, but it doesn't get anywhere near as silly as shows like Bo-Bobo or MAR. While the series is linked to Slayers, it is by no means required to follow the show. Obviously, the first volume is a great place to get started in this series, so if you've heard of the show and wanted to get into it, or if you are looking for a new comical anime series, give this one a look.