Hi I dont know exactly in what period you are interested or how much detail you are interested in going into but the following three books are useful introductions.
Robin Osborne, Greece in the Making, 1200-479 BC. Second edition (first edition published 1996). Routledge History of the Ancient World. London/New York: Routledge, 2009. xx, 377. ISBN 9780415469913 $41.95 (pb).
Simon Hornblower, The Greek World, 479-323 BC. Fourth Edition (first published 1983). Routledge History of the Ancient World. Abingdon; New York: Routledge, 2011. xxi, 410. ISBN 9780415602921 $42.95.
Graham Shipley, The Greek world after Alexander, 323-30 B.C.. Routledge history of the ancient world. London and New York: Routledge, 2000. xxxi, 568 pages : illustrations ; 25 cm.. ISBN 0415046173 $29.95 (pb).
Reviews are here:
https://bmcr.brynmawr.edu/2009/2009.06.37/
https://bmcr.brynmawr.edu/2011/2011.12.46/
https://bmcr.brynmawr.edu/2001/2001.03.11/
Here are some other more general possibilities culled from a UK university web site:
Beard, M. and Henderson, J. Classics: A Very Short Introduction (Oxford 1995)
Boardman, J., Griffin, J., & Murray, O. The Oxford History of the Classical World (Oxford 1986)
Cartledge, P. The Greeks (Cambridge 1993)
Easterling, P. & Knox, B. The Cambridge History of Classical Literature (Cambridge 1985)
Hall, J. M. A History of the Archaic Greek World c. 1200 - 479 BCE (London 2006)
Murray, O. Early Greece (2nd ed. London 1993)
Neer, R. Art and Archaeology of the Greek world: a new history 2500 BC-150 BC (London 2012)
Vernant, J.-P.The Greeks (Chicago 1995)
Vernant is always a very interesting author to read.
You mentioned Pomeroy, A brief history of Ancient Greece which I haven’t read. A review is here
https://bmcr.brynmawr.edu/2005/2005.02.06/
The review mentions Amos and Lang (which again I haven’t read) and notes " … [it]has not been revised and does not reflect recent work." This may or may not bother you.
I also think reading Greek historians is important but this won’t do on it’s own. Most of our knowledge of pre Classical Greece (and a good deal thereafter) comes not from those texts but from archaeology and inscriptions.
Finally there is the venerable but out of date and sometimes wrong account “A history of Greece” by Bury and revised in 1973 by Meiggs. It is a serviceable account if you are mainly interested in who did what when. Its not the approach taken by contemporary scholarship but may appeal.
From this small collection of suggestions you can perhaps see that the range of possibilities is enormous. I hope you have a local library (or bookshop ) where you can dip into these and choose what best meets your needs.