My academic background is in Greek history and literature. Even though my duties often require me to spend time elsewhere, I make sure to devote as much time as I can each day to keeping up with my field. That means I tend to be the go-to guy at work for questions about all things Greek (We have a couple others that fill that role as well). When I'm reading, then, I try to keep an eye out for things that would be helpful to a beginning student of the Ancient Greeks. Below are some books I've found helpful over the years as first steps in beginning to understand the Greeks and their literature.
For a basic history of Ancient Greece, I recommend starting with Thomas R. Martin's Ancient Greece: From Prehistoric to Hellenistic Times. Ancient Greece walks you through the development of Ancient Greek culture in chronological order and alerts first-time students to the major points of interest. Martin's book should be supplemented with The Oxford Illustrated History of Ancient Greece and the Hellenistic World by John Boardman et al. This work covers the same time periods using a topical (as opposed to narrative/chronological) approach that serves as a starting point for understanding the key issues in Ancient Greek and Hellenistic history. As a useful and basic introduction to Ancient Greek, Hellenistic, and Roman thought, particularly philosophic thought, I recommend When Athens Met Jerusalem by John-Mark Reynolds.
When moving into the literature of the Ancient Greeks, I like to begin with the Cambridge Companion series. The Cambridge Companions are topical and meant to serve as starting points for the major areas of academic interest in a work or set of works. The areas dealt with will differ slightly from work (or group of works) to work, but usually include: author(s), composition, language, narrative, background, social issues, and reception. Also helpful, are the Blackwell Companions to the Ancient World. Like the Cambridge Companions, they are topical, but the Blackwell Companions often (though not always) survey larger topics such as "Ancient Epic" or "Greek and Roman Historiography." They can also be helpful in surveying the key issues of a particular historical period or society (ie. "Sparta," or "Late Antiquity"). The suggested readings and bibliography sections in these works are particularly useful for putting together a plan for more in-depth study on a particular topic.
Buying even just a few of these volumes can become an expensive endeavor, so I do recommend making full use of Inter-library loan before you buy. Remember, many libraries, particularly in large cities or university towns, allow patrons to apply for a special card that grants limited access to university and state college libraries. One important resource that students of the Ancient World should be aware of is the Loeb Classical Library of Greek and Roman texts many (though not all) of which have recently come into the public domain.