K-On! is an anime based on four (eventually five) high school girls who perform light music, the literal translation of K-on, band. As expected, it's driven by the girls' music. There is not much else to the show, to be honest. There is some comedy, but the jokes are far apart. Of course, people aren't watching this to laugh: they're watching it to look at cute things and hear cute music. This is where K-On! and K-On!! ultimately succeed. Its animation style and music really deliver, and by "really deliver," I mean it is incredibly adorable. It's clear that the creators were greatly influenced by Japanese music as the characters of the show are all named after members of Japanese bands like the The Pillows. Don't dismiss this is as the musings of an otaku (please haha), it's true. K-On!'s music has no real substance, but it's nice to hear, much like the rest of the show.
It's good that the show doesn't take it self completely seriously; if it did, then there would be a problem. To wrap things up, K-On! and K-On!! are fun to watch, but not an appropriate choice for someone looking for a deep, intellectual show. It's lighthearted, mindless fun.