What are the best sources for learning about the Greek and Roman Empires?

To begin, I will ask what exactly is it you refer to when you say Greek Empire? With consideration to the fact that what we term Greece was, in antiquity, not a united political system; this term can therefore refer to numerous empires: the Macedonian/Alexandrian, the Successor Kingdoms, and even the Delian League according to some (a number of historians have taken to terming it the Athenian Empire.) If I am to presume, until you specify which it is you refer to, that you wish to seek sources for the study of the entirety of Greek antiquity and the Roman Empire then I possess another question: what type of sources do you seek? Primary sources? Secondary? (I'm not criticizing your post at all by the way, rather I wish to be specific so as to be of the most assistance as possible; I love seeing more people interested in our antiquated legacies!) If you're searching for a particular form of sources and/or specific periods or regions, please do comment and say as such! For now I advise as follows: let us presume that you are in the beginning phases of your study of antiquity; I suggest, specifically for Greece, the book "Classical Greece" by CM Bowra. The book is fairly short (circa 120 pages) and provides a general overview of various aspects of Greek society, influence, politics and other history; it is great as a short introduction to the subject. It will lend to you a general sense of the timeline of Greek antiquity. Perhaps you elect to seek a further, more comprehensive and serious study? For Greece and Rome alike, there are many books to be had on the subject! One which may interest you is "Egypt, Greece, and Rome: Civilizations of the Ancient Mediterranean" by Charles Freeman; it's much more lengthy than the book previously described, as well as it succeeds in interrelating the 3 civilizations: a theme which will reoccur often in your studies. We should not limit ourselves to these however; primary sources, or sources which originate from the men and women whom were actually living during the period, are invaluable. You can find these sources in a number of areas: check your local libraries perhaps; many of these sources may be found online through various sites; contact universities which host a classics department and speak with them in regards to locating primary sources and/or reviewing ones they may store in their libraries. With honesty, there are a multitude of sources to choose from: scholarly journals, numerous books, primary sources, classes taught on the subject, etc. I suppose it is dependent upon how serious you are in your studies. There a multiple primary sources I could cite for you: from Plutarch to Thucydides and beyond. Furthermore I can recommend several books, both in detail and as introductions. You may always check with your local libraries for books on antiquity as well. I hope this has been helpful; if your question was intended for specific sources rather than forms of sources as well as the whereabouts at which you may locate said sources, please do comment. I would be more than glad to assist you via a providing of multiple specific sources both primary and secondary!