I did not grow up in a household where art was a part of daily life. In fact, I didn't see much of any kind of art except in books in college.
I can remember being awestruck the first time I saw a real piece of Greek pottery. I was in the presence of history. No 2-D picture in a book could begin to evoke the feeling I had seeing that pot in person.
I've had many of those experiences since that first one 50+ years ago. One was the first time I saw Michelangelo's Pietà at St. Peter's Basilica. I'm neither Catholic nor religious, but to me it remains the most moving and beautiful piece of sculpture I have seen.
I loved that article about medical students studying art to learn about what to look for in their patients. Much of what we know about history before the invention of the camera is through art. That's the only way we can see what people looked like, how they dressed, how they lived, where they lived and what was important to them, as shown by the artist.
There are many excellent general resources for learning about art. I just ordered a DVD of Nigel Spivey's How Art Made the World: How Humans Made Art and Art Made Us Human. This was a BBC series from the mid-2000's. I can hardly wait to watch it.
Googling produced lots of resources about it including the DVD, the book and even a way to watch the episodes online. Perhaps others who've seen the series or read the book can jump in here about it.
Rick Steves has a couple of useful books on European art, Europe 101: History & Art for the Traveler and Europe's Top 100 Masterpieces: Art for the Traveler.
As much as I learn and love it, seeing art is exhausting. Part of the exhaustion comes from the high level of visual input. Being highly selective helps. Taking a break in the coffee shop helps. Sitting on a bench staring at a particular work helps. You'll find what's best for you.
Guides that concentrate on a small number of highly important pieces are one way to be selective. Exploring more online about what's in a museum can help with targeting what you want to see.
One thing that works for me is to pause at the entrance of a gallery and scan what's there. If nothing compels me to take a closer look, I move on. There are no rules that require following the herd around the perimeter looking at every picture and reading every label. There is no requirement that you see everything in the museum.
About David. The last time I was in Florence, I limited my Accademia time to only the Michelangelo sculptures. I see things pretty quickly. I'd guess I spent 30-45 minutes tops.
This link to a search for them shows some of the pieces others have mentioned. As I remember, all of them lead up to David at the end of a long hall-like gallery. And he is awe-inspiring. This is the link to the story about him on the Accademia website.