DMR is overall better, but I would recommend to get just as familiar, if not more, with the BR.
I'm curious. Why get familiar with the BR? I'm not opposed to the idea just because I like it as a weapon and choose it when I play SWAT, but ... why more familiar with the BR than the DMR even though you admit DMR is the better weapon?
Because it is the starting weapon in competitive settings modes. Although they are only available in custom games at the moment.
Doesn't really answer it, though. I mean, unless your goal is to become a Halo pro, which doesn't exist anymore, thanks to MS and Virgin's contract precluding MLG involvement, that doesn't mean anything. Besides, the Superman answer was already pretty good, but "competitive recreational activity" is just as funny to me.