Does anyone have recommendations for how best to approach learning Ancient Greek? Would this book make a good starting point?

I’ve found this “First Greek Book” [1] to be a good starting resource. It gets you through Xenophon’s Anabasis which is a great starting book.

1: https://daedalus.umkc.edu/FirstGreekBook/

I’m a fan of self directed learning, but if you can find a class, say at a local adult education system, it will likely make a huge difference. They are not mutually exclusive, and in fact having self-studied will make things the teacher says more likely to make sense when you first hear them.

There are some things in Greek like the passive periphrastic which are essentially unknown in American English (similar constructions are more common in Australian or Hindi) and which are probably very hard to understand without some discussion.

Likewise the Ancient Greek culture is profoundly weird and nothing like it’s described in popular culture today. The society depicted in The Odyssey is in many ways much closer to ISIS than what you might expect and again, discussion will make Greek literature much more visceral and can unlock the pleasure of some of the classics.

This covers maybe the first 1% of the effort, but Greek Script Hacking[0] is quite excellent.

0: https://www.amazon.com/Greek-Script-Hacking-optimal-alphabet...

I recently read this book in English translation. It was a fun and informative read, though frankly I don't know how much I really retained.