• Aria
  • pure slice of life, heartwarming
  • start with Aria the Animation, NOT Origination despite the name
• Hidamari Sketch
  • pure slice of life, art school, comedy
• Nichijou
  • comedy, school
• Danshi Koukousei no Nichijou
  • comedy, school
• Mushishi
  • supernatural, travel
• Lucky Star
  • comedy, school
• Minami-ke
  • comedy, family

If you especially liked K-On!, then I guess Lucky Star is the closest one, with no music band stuff. Note that LS has tons of references to other media, so if you're new to anime, 80% of the jokes will fly over your head.

Nichijou is very comedic, and so is the other one with Nichijou in the name (they are unrelated, just the generic term "nichijou" means "daily life").

The Aria series is my favorite; it's very slow, to the point of being boring, but you grow to really love the characters and end up bawling your eyes out when it ends.