The books that come to mind to me are somewhat outdated, so not everything is as they lay it out, but are really well written and oriented towards a non-specialist audience.
First, I'd recommend The Bull of Minos, by Leonard Cottrell. He is a journalist, so he writes well. The book gives a brief introduction to the several civilizations that developed around the Aegean sea in the centuries before classical Greek civilization.
A great book on the period itself would be The Life of Greece by Will Durant. Also very well written, it attempts to give you an understanding of social mores, literature, poetry, art, philosophy and other fields in addition to the great wars that marked the period. It may seem rather large, but it is very welcoming to non-specialists who are just curious about Greece. Again, the book is more than a few decades old so not everything is as up to date as it could be, but I'm recommending it as a starting place given how well written it is as well as the scope (ie. it isn't just a narrative about a bunch of battles).
Once you have that down I'd really recommend reading some actual Greeks. Yes, you can and should read histories by specialists as you learn more. However, you should now have the basis to understand what's going on in original materials, like plays and histories. I'd strongly recommend reading Herodotus' Histories followed by Thucydides' The Peloponnesian War.
On a personal note, The Peloponnesian War reads as cut throat as any A Song of Ice and Fire novel and good translations are a pleasure to read. I remember getting to one part 'and thus ended the Sicilian expedition' and having to put the book down, walk to the balcony and mutter 'daaamn.'