Hey there!

I'm writing this as a PhD in Roman history. Thus, this is just opinion but with some explanation behind it. I can give detailed comments only on the period through the Punic Wars.

The order you approach these texts really depends on what you're looking to get out of them. If you just want to learn a bit about Rome, you really can't go wrong with any order (except maybe leaving Cassius Dio for last). However, if you want to develop a strong understanding of Rome, the best approach that only involves the ancient sources is probably topically chronologically. That is the approach I have taken in my recommendations (along with recommended translations).

To start: Livy books 1-5. This provides a foundation of knowledge about the monarchy and early Republic from mythical times to the Gallic Sack (387 BCE). The translation I have linked is the best in English, and there is a short—but interesting—introduction. Accompany this with Plutarch's early Roman lives, collected in an efficient Penguin Classics volume. The first five lives are all figures that you will have read about in Livy. I would recommend reading the Plutarch biographies after reading the relevant sections in Livy.

Next: Livy books 6-10. The second set of five books of Livy takes you from the Gallic Sack to the Battle of Sentinum. This covers many aspects of the Roman conquest of the Italian peninsula, but unfortunately does not give us a history of it all. Livy's work after book 10 (to 20) is lost to us in all but excerpts. I would recommend filling the gap between this and the period of Roman overseas expansion in three ways. First, you can read the Periochae of Livy which are short summaries of each book, including those otherwise lost to us. Second, Kathryn Lomas' recent The Rise of Rome provides a great overview of the Roman expansion into Italy that also includes archaeological evidence. Third, you can read the fragmentary work of Cassius Dio, with some of this period preserved.

Next: Polybius book 1. Polybius is another ancient historian whose work is not preserved in its entirety. The first book, however, provides the primary narrative for the First Punic War. Supplement this with relevant sections from Cassius Dio.

Next: Livy books 21-30. These cover the Second Punic War. Read this alongside the next five lives in the above cited Plutarch collection (Fabius to Flamininus). Those lives will provide a preview of the next decade of Livy, as well. You can supplement these with the relevant sections of Appian (a historian not on your list). Supplement this with relevant sections from Cassius Dio.

Next: Livy books 31-40, and 41-45. These mostly focus on the expansion of the Roman Empire and wars in the eastern Mediterranean. Supplement this with relevant sections from Cassius Dio.

Unfortunately, this is where my expertise ends. I can't give any more solid advice than to read Suetonius and Tacitus in their entireties. But, add a few things to these for a fuller understanding: Caesar (both the Gallic War and the Civil War) and Sallust (both Cataline's War and The Jugurthine War). This will give you at least a foundation in the late Republic, though there is considerably more to read there.

I also have two pieces of general advice. First, though not specifically about Roman history, reading Diodorus Siculus will give you a good overview of the Mediterranean world and contains some interesting tidbits of early-ish Roman history. Second, grab some of the modern surveys about ancient Rome. I highly recommend The Edinburgh History of Ancient Rome series. These books are all up-to-date and accessible to general readers, students, and scholars. Given the fragmentary nature of Roman history, this approach will give you a more coherent view of ancient Rome than reading the primary sources on their own. These books also incorporate archaeological evidence which is of the utmost importance for trying to understand Rome.

If you have questions, feel free to reach out via DM. I'm always happy to help people connect to Roman history!

- Josh