I would recommend an 2004 turn-based strategy Legion from Slitherine (currently it's available either on Steam or on Slitherine website, I would recommend the latter). Very accessible and aside from very solid core, has some nice details, like your army on the world map answering to you in Latin (if you're playing as Rome of course)

I would recommend Caesar III and IV for a good Roman experience. Caesar IV appears to be particularly well researched and designed. I suggest this as a Latin scholar and Roman historian. It also teaches much about Roman life in a realistic setting.

Added vote for Caesar 3, one of the city builders that have stood the test of time and is still great to play nowadays, it's just my favorite city builder of all the Impressions games and i continue to have some playtime with it every few months since decades
the roman atmosphere is really there with the soundtrack being always in a corner of my mind .

Give a look to the Augustus fan expansion that add a lot more settings and content to build with to get even more game time from that already fantastic game.

Hi, you have confessed to me; I mean convinced me of Rome: Total War.
After all, Rome wasn't a funny place, and that's why they fell. But before they kicked the bucket some stories for the movies were given. Having that game since today.

Enjoy! :D

You should publish your reactions to those games, too. Perhaps you will not enjoy R:TW or the others as much as I would.