I would also recommend any ancient history written by Tom Holland - "Persian Fire" about the Persian invasion of Greece is a good introduction to the era. Also, Adam Nicholson's "The Might Dead: Why Homer Matters" is a brilliant read, linking such disparate topics as Albanian and Scottish bardic traditions, and LA gang culture.
I'll second Tom Holland. I read "Rubicon" not too long ago. I would recommend pairing it with Dan Carlin's Hardcore History podcast series on the era for an extremely cohesive and engaging account of the Roman republic's decline.
Loved Persian Fire, in the middle of Rubicon right now as well. Excellent call on the Hardcore History - King of Kings three part (13 hour!) podcast. Listened to this falling asleep in my hammock on a backpacking trip for a week, amazing.
The ancient Greeks and Romans are fascinating histories. I've always thought of the Greeks as philosophers and the Romans as engineers. In fact I learned to love both taking Latin for 4 years in high school because the teacher loved to take a whole week to teach us history whenever someone asked him a question about something in old Rome.
