Reference Book on Greek & Roman Society

The encyclopedic Oxford Classical Dictionary is the gold standard. It covers just about everything under the sun: politics, people, places, technology, battles, literature, philosophy, etc. You can usually pick up a used one for about $50. Make sure you get the revised edition, edited by Hornblower, as it is much more up to date and reflect current scholarship.

H.H. Scullard - "a history of the Roman world", Routledge. It's divided up into two books - 753-164 BC and "From the Gracchi to Nero". It does speak about the Greeks and Diadochi, but from a Roman perspective. I found it fascinating all round. Then again there's Robin Lane Fox's "the classical world", which is an excellent read, the cambride companions to various areas of history and of course, the sources themselves - have a read of Herodotus 1-4 for a general "feel" for Greece, read some Greek tragedies for information about morals, read books V-VII of Plato's republic, Polybius's histories (any bits that take your fancy) and then read some Titus Livy. That will give you a good overview from primary sources. It's really very hard to answer this question - I think one just has to copiously read as much as they can to get the general idea - I have no classical education (well I've started a classics course recently at college) yet I've gained my (admittedly quite good) knowledge of the period through just reading as much as I can. My starter for 10 would be don't neglect primary sources in this endevour of yours.