There is not a good comparison in English for Greek dialects … it depends on which dialects you are comparing. Let me give heavily simplified description of the relationship of the Greek dialects -
The two super-groups are Doric and Aeloic-Ionic
Aeolic-Ionic splits into Aelolic and Ionic
The Epic dialect of Homer (aka Old Ionic) is primarily an Ionic dialect, but it has some archaic forms not used frequently in later dialects, as well as having a dose of Aeolic thrown in.
Ionic dumped its archaic forms (except in poetry), and the grammar shifted a little (for example, the demostrative became a definitive article), but it is mutually comprehensible with Homer (after all, they still enjoyed the epics in the original language).
Attic split from Ionic, so Attic and Ionic are pretty close. It might still be useful to look at a short article comparing the dialects before you make the jump from Attic to Ionic (or Ionic to Attic), but the jump should be pretty easy.
The Doric dialect is more distant than the other dialects however in the canon of Greek literature you will only find Doric blended with other dialects. Almost anything with Doric will also have a strong Epic dialect influence (actually, almost all poetry has a strong Epic influence). Sometimes you will also find some Aeolic. In the case of the Athenian dramatists, Doric is mixed with Attic, but since the composers are Athenians Attic is the main dialect at work. Therefore, you should probably be aware of Doricisms, but since it’s almost always adulterated you really don’t have to learn the full dialect.
Koine is descended from Attic Greek. I do not know how different they are, but the leap from Attic to Koine is supposed to be very easy. The leap from Koine to Attic is supposed to be harder.
One of the trickiest things is the change in vocabulary - either words fall out of use, new words come into use, or (beware) words change meaning. This is where dialect-specific lexicons are nifty. However, the only dialects worth investing in a lexicon for are Epic, Attic, and Koine.