I know more about empire than early republic, but as far as I know Polybius provides pretty much a unique insight into the Greek and Roman worlds of that period, due to his interesting perspective of being a Greek now on the "Roman side" as it were. Fortunately for you, the nature of that era and the following 2 or 3 centuries was one of literature, and we have a very rich set of texts, covering all sorts of things. Below I'm just putting 'Histories', but if you want to branch out into farming, water, architecture, poetry etc., you can.
I don't know as much about the Greek side of things, but I guess Arrian's
Anabasis of Alexander the Great might be of interest to you, not something I've read though.
Appian's
Bellum Civile is worth reading, covering the slightly later period, and the
Lives of Plutarch make very interesting reading. I'd say the latter would be your best bet, covering a wide date range in very interesting ways. Bear in mind he was writing in the 2nd century AD though.
Livy's
Ad Urbe Condita (or so it is often called, sometimes just
Histories) is fragmentary in parts, some bits only known to us through what's called the
Epitome, a later summary of the work. It covers the history of Rome from 753 BC up to the 1st century BC in a logical way, easy to read for someone without too much knowledge of the times.
Sallust's
Bellum Jugurthinum,
Bellum Catilinae and very fragmentary
Histories cover the last century BC, and Caesar's
Gallic Wars and
Civil War are relatively easy and fun to read in my opinion, looking at the end of the 1st century BC.
In terms of succeeding things, Velleius Paterculus' history isn't widely known outside of those knowledgeable about the subject, and it must be read with some caution and knowledge of its writer. Tacitus provides what is probably the most complete and closest-at-hand account of the first century AD, with his
Annals and
Histories. The Annalistic basis might make it quite hard to read if you're not that interested in the details, but if read in conjunction with a good history book/wikipedia you should enjoy it. Suetonius'
Lives of the Caesars is a wonderful gossipy set of works on the first few emperors, court intrigue and affairs and the like, very opinionated (not that Tacitus isn't) but easier to read.
Later still you have Cassius Dio's
Histories, ritten in the 2nd century AD, very long, fragmented in parts, attempt at history of Imperial Rome. Borrows heavily from Tacitus and Suetonius and other authors, gets some things wrong, gets others right.
Later still you have Ammianus Marcellinus'
Histories, an attempt at continuing Tacitus. Unfortunately lots of bits lost to us, but still a fair amount to read if you're interested in the later Roman empire, 3rd century AD onwards and the shift to Christianity.
With any of these texts though, bear in mind that you mustn't treat them as true, unbiased history (if such a thing is possible). They were written with purposes in mind, and in a time when the idea of history was not the same as it is now. But enjoy
Hope this is helpful. There's a lot of material here, should take you a few months/years
If you need more though this is far from everything
