Hi Yeti, I'd start with a general book called
"Heroes, Gods, and Monsters" by Evslin. The book is aimed at younger readers but in my opinion it's a more fun/accessible place to start than Hamilton. It's definitely worth the two days it will take you to read it, and it's the book to give as a gift to some bright eleven-year-old that you think might like mythology.
But then, yeah, I'd read Edith Hamilton's Mythology for another general survey. By now lots of the names and relationships will be sorting themselves out in your head. Hamilton gives pretty faithful translations to the actual texts, but she cuts out boring stuff for the sake of storytelling. Read the introductory chapters too, they give good context.
After you've read those then you'll have definitely have an idea of the specific characters and stories you like, and then you're good to go. Before buying something you should always thumb through a few translations for a minute to see if you're responding to one better--some of the "standard" translations were done two centuries ago and sound stilted to a lot of people today. If you want to read Homer, you might think The Odyssey is more fun and less confusing to start with; I'd rate the translations 1. fagels 2. fitzgerald 3. lattimore, but whatever. (For The Iliad I'd go 1. lattimore 2. fagles 3. fitzgerald.)
If you want to read about some of the dramatic patterns that underlie all mythology/stories, you might like Joseph Campbell's "The Hero with a Thousand Faces." Also Joseph Campbell's interview series with Bill Moyers on PBS was terrific.