I have heard many folks say that a contemporary Latin speaker would have little trouble understanding Classical Latin, and was wondering if it was the same for Greek. Also, how about a contemporary Greek reading Classical Greek?

The two situation aren't analogous. Contemporary Latin speakers are analogous to "Contemporary Classical Greek speakers", as in, they are people who speak a dead language (nobody's native language is "classical Greek"). The analogous situation to a "Contemporary Greek speaker" (or "Neo-Greek speaker") is a "contemporary Romance language speaker", because contemporary Greek is Classical Greek with centuries worth of language change, same as Italian, French, Catalan and Romanian are Latin with centuries of change.

Also, posting this question to /r/linguistics might get you even better answers.