Before you can answer the question "which Greek dialect should I study first" you need to know the answer to "what do I want to read in Greek." Unless you're studying at a school, where you're unlikely to have a choice in the matter, your intended reading matter determines the best plan of study.
If you're going to read only the NT and related Koine works, then you should probably study Koine.
If you plan to read <i>any other Greek</i> at all, I strongly recommend starting with Homeric. All later literature references Homer to some degree. Even if you plan to read only philosophy, the Greek philosophers like to quote Homer, too.
That's the very short answer to the question.
For a fuller discussion about all the literary dialects of Greek, please see Greek Dialects: Where to Start? There I discuss in more detail the pros and cons of the possible starting places, and also touch on some of the obscurer literary dialects you might run across.