If you are going to read Greek classics I recommend the Landmark series, simply because of all the extra material and maps etc. So far they have the Landmark Arrian: The Campaigns of Alexander, Landmark Julius Caesar: The Complete Works; Landmark Herodotus: The Histories, Landmark Thucydides: Guide to the Peloponnesian War; Landmark Xenophon's Hellenika; and Landmark: Xenophon's Anabasis.
I'm waiting for the Landmark Polybius: The Histories (if they ever get around to it).
There is also The Library of Greek Mythology by Apollodorus, which is something like a Greek mythology encyclopedia.
Constellation Myths by Eratosthenes can be considered an extension of the above, since it is also in semi-encyclopedia format.
Ovid's Metamorphoses also provides extensive details about Greek mythology There are various verse translations (Rolphe Humphries and Horace Gregory translation seem perfectly understandable to me) but also a prose translation by Mary M Innes, which reads just fine and is fairly accurate as far as I can tell.
There is the usual Illiad and Odyssey by Homer, as well as the Aeneid by Virgil (more a Roman thing?) Incidentally, I found a Macmillan Collector's Library edition of the Odyssey translated by "Lawrence of Arabia" in a used book shop recently. I thought that might be interesting to compare to my Fagles translations.
But I also enjoyed Voyage of Argo (aka Jason and the Argonauts aka the Argonautica) by Apollonius of Rhodes.
Then the collection of Greek Plays by Aeschylus, Sophocles, Euripedes, and Aristophanes.
OFF TOPIC: If you want a book that provides an analysis of Celtic myths (it's not a story book) then James MacKillp's Myths and Legends of the Celts is very useful.