The best place to start reading Archie comics, by far, is the new 2015 series. This was the rebirth of Archie comics into a modern era, and they could not have gotten a better team to do it. With one of the best first issues I've ever read, I guarantee you will be sold.
Here is all of Archie (2015) in reading order:
There is also an incredible universe of other books being built up around this series. Especially the hilarious Jughead: