Don't listen to those other kids, this one is right:
Oh for heavens sake . . .
Both are used to say thank you in Japan.
Domo is usually used as a quick little thank you to something like maybe holding a door open or something. Where as Arigatou has a stronger connotation, and is definately used in more formal contexts than Domo. Heck, even the phrase Domo Arigatou is used, which is similar to Arigatou Gozaimasu (thank you very much).