Like TheLastDays Rome was also the game that introduced me to the series. I think nothing could go wrong if you start with Rome; plus there are lots of awesome guides out there e.g. Frogbeastegg, Smackus Maximus etc. that could make you an expert in no time. Medieval II is good too, but I would maybe leave Empire till a bit later. It requires Steam to play and asks a lot from your computer; furthermore the campaign map is vastly different from the other two, and the battles are also different, with more focus on gunpowder instead of swords and bows and arrows.

Welcome to the Org TweedyTodd.