The previous post suggests that, leaving aside those who want to read New Testament Greek exclusively, everyone should start with Homeric Greek, even if their main interest is philosophical Greek. I think this idea is utterly misguided and wrong-headed. It’s much easier to pick up Homeric Greek after learning Attic Greek than the other way around.
A large part of the difficulty in learning Attic Greek lies in mastering the complex syntax. Homeric syntax is much simpler. You’re better off learning Attic syntax first and then picking up Homeric Greek, especially if your ultimate aim is to read philosophical Greek and other Greek prose. If you learn Homeric Greek first, basically you almost have to start all over again from the beginning to learn Attic Greek.
The difficulties of Homeric Greek, in contrast, lie in the morphology, and to a lesser extent, the vocabulary. The Homeric poems exhibit a large array of forms and words drawn from several dialects–in many cases, there are alternative forms of the same words from different dialects to fit different metrical slots. It’s easier to pick up and recognize the proliferating forms against the stable (more or less) normative background of Attic Greek, than to try to assimilate all of the alternatives at the beginning stage of your studies. And, while it takes some effort at first, once you get going with Homer, the strange forms and vocabulary shouldn’t pose as much of a problem as they seem to at first if you have the normative background of Attic Greek at your disposal.
Homeric phonology is not too different from Attic. The most salient difference is η after ε, ι, and ρ (Homeric) instead of α (Attic).
In sum, unless your sole interest in learning ancient Greek is to read the Homeric (and you should bear in mind that as you proceed in your studies, your aims may evolve once you’ve read the Homeric poems), you’re better off, in my opinion, starting with Attic.
I should add, however, that “Homeric Greek” is something of a misnomer. In general, all Greek hexameter and elegy from the archaic period down to Quintus Smyrnaeus and Nonnus in late antiquity was de rigueur written in the “Homeric” language. (The exception: in the Hellenistic era, Callimachus and Theocritus wrote hexameter poetry in a literary Doric language.) So Homeric Greek will give you access to a larger body of poetry than is implied by the adjective “Homeric.” But dramatic dialogue is written in Attic and dramatic choruses are written in a lightly Doricized version of Attic, so if you’re interested in reading tragedy and comedy, you need Attic.
As for the other literary dialects–Ionic (very close to Attic), Doric and Aeolic (a very limited corpus consisting of largely fragmentary poems)–once you have Attic down, these won’t present much of a problem. (The emphasis is on “literary”; actual spoken dialects, as evidenced by local inscriptions, diverged much more sharply from one another and from literary Attic.) Anyone who wants to read the full range of ancient Greek literature has to be prepared to encounter an array of dialects, but with the normative background of Attic, they shouldn’t have too much trouble with other dialects.