The best place to start reading Archie 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.
Here is all of Archie (2015) in reading order:
If you're more into comedy than drama though, you may prefer Jughead. There are also two great series, Sabrina and Afterlife with Archie, which are fantastic horror books: