The most obvious change is that the letter beta has shifted from a B to a V sound.

There were lots of little changes in idiom. My classics proff said he could see that the hotel clerks could understand him but were trying to keep a straight face and not laugh at his archaic speech.

Some ancient dialects are unintelligible to the modern ear. Modern Greek is derived from Attic/Ionic Greek, the classical prestige dialect of Athens. The only other dialect still surviving is Doric Greek, which is still spoken by a few hundred people.

So a lot of ancient Greek would sound a bit gibberish.

Dr. Ioanna Sitaridou is most likely the leading person in this subject. In northern Turkey there is a region called Romeyka, which speaks a nearly unchanged version of Ancient Greek. Here's a quick video on it.

This link is the Hippocratic Oath in Ancient Greek. Compare to Modern Greek.

Britannica has a pretty good write-up on the differences.

Side question - how would we know what ancient languages sound like? What method would be used?

No. I have taken Classical Greek, which is usually one particular variety whose name I've forgotten. Koine I think? While of course we can't guarantee that we're speaking it completely the same, there are general principles about its use. I've also heard and read modern Greek, and basically a lot of abbreviation and simplification has happened. I haven't made an intense study of this, so I can't expound on that too much, but, for example, Gamma in Classical Greek is usually more of a soft g sound, and syllables are uttered distinctly, whereas in modern Greek, Gamma is more of a y sound and syllables are more run together.

For example, the word that transliterates as georgas (literally earth worker, meaning farmer, and root for the name George, sorry, I don't know off-hand how to produce Greek letters on a Qwerty keyboard) is pronounced gay-or-GAHS in Classical Greek and YOR-gus in modern Greek, though it is pronounced slightly differently depending on whether you're talking about a farmer or someone named George.