Thanks to a Destructoid post, I'm kinda obsessing over this old anime.
The anime is called "Honey the Bug." It came out in 1986, and is based on the game Adventure Island on the Famicom. It went on for 51 episodes, got a movie, and it's own game. From the drawing style and the character interactions in the intro alone, it really seems like a kid's show, but for some reason I want to find out more about it. I like the simple animation style used for the show. Unfortunately, you can really only find the intro and endings to the show; I haven't found a full episode source yet.
If anyone can get me some more info on this, or some clips (even better), please feel free to post. Here's some of the videogame footage, for both Adventure Island and Honey the Bee.