I started with Edith Hamilton's book Mythology. I then read Bulfinch's Mythology.
These two books are a great primer. Many people also like Graves' The Greek Myths, but I find it to be a bit mired in detail.
After that, reading the classics themselves is a must. Start with Homer's Illiad and Odyssey. I like the Fitzgerald translations myself.
Then read Hesiod's Theogony and Works and Days. (two short works).
The Greek tragedians, Aeschylus, Sophocles and Euripides are essential.
There are quite a number of plays by these guys, but at least read the Oresteia trilogy and the Oedepus trilogy. Lattimore's translations are good.
Then move on to The Voyage of Argo by Appollonius of Rhodes.
Ovid's Metamorpheses is a great compendium of Greaco-Roman myths. I loved the Gregory trans.
Finally, read The Aeneid by Virgil, although it is more to do with Roman myth than Greek myths.
There is still much more than this, but I feel it is a good, well-rounded place to start.
The above are merely my suggestions, others might feel very differently on where to start.
Also, the above does not include much in the way of the interpretation of the myths; it is more to familiarize yourself with the
myths themselves.
Happy reading!