1. The grammar to get is Smyth. It's very long and has more detail than you'll probably ever need, which makes it the ultimate reference grammar. For quick reference, something like the Oxford Greek Grammar will do, but it offers nowhere near the breadth of Smyth.
2. The biggest difference between the different dialects is essentially spelling. Imagine that English speakers spelled every word exactly as they pronounced it. Now, imagine what the differences would be between British, American and Australian spelling - this is about what the difference between the different Greek dialects. There are also a few inflectional endings that are a little different, but nothing that can't be learnt fairly easily.
The major dialects you will need to know are Attic (in which are written the tragic and comic plays, most philosophical works, and mostly everything written after the fifth century), Doric (which playwrights use to write choral passages), Ionic (in which Herodotus wrote his historical work, and of which Homeric Greek is a variation), and Aeloic (in which a few poetic works are written).
New Testament Greek, aka Koine Greek, is a simplified version of Attic Greek. Athens was so prominent in the literary world that after the fifth century, all Greek was modeled after it. Learn Attic Greek, read some Plato and some Xenophon, and marvel at how much easier New Testament Greek is. The forms are simplified, and irregularities are smoothed out.
I learned Greek with Mastronarde, and I liked how thorough it was, but many of my colleagues disagree. I would recommend it or Hansen and Quinn, and I would recommend taking plenty of time to do each lesson in full, memorize all forms completely, and know as much vocabulary as you can cram in your head. If you already know Latin, you'll be surprised at how many more verbal forms Greek has, and at how much larger its vocabulary is. But it's a worthwhile pursuit!