Where: Sonoma, Thunder Hill, Laguna Seca are your closest tracks.
Budget: Very dependent on you. Cheaper track days will have more cars in a session and (in my experience) tend to attract more reckless drivers. I've seen parts fall off cars, people brake check me trying to give me a point by that I was waving off, and other dangerous things that aren't worth the lower cost. If it's you and your son's first track day, I highly recommend a track day with an instructor. I know Hooked on Driving and Audi Club Golden Gate have great instructors. It might get a little tricky if both you and your son are novices, since you'd be in the same group. Audi Club has an event at Thunder Hill next month, maybe they'll let you drive one day and your son the other?
In terms of consumables, you'll take a decent amount of life from your brake pads and tires. If this is going to become a consistent thing, you'll eventually want some track pads and tires. The R in stock form is fairly capable on track, the only thing I would say is a requirement prior to going is a high performance brake fluid. RBF600 is a common fluid used.
Add in helmet rental if you don't have your own, a hotel depending on how far of a drive it is, and general food costs.
Insurance: This is dependent on your risk tolerance. As previous poster mentioned, normal auto insurance does not cover track days. If you can't afford to walk away from the car if you crash it, get insurance. I, personally, have never bought track insurance but that's my risk. My friend totaled his TTRS at a track day, and he was very happy he had track insurance.
Connect with some guys on the forums about track days in the Bay. I've met a good chunk of them and they're all great guys.
https://www.golfmk7.com/forums/index.php?threads/bay-area-track-days.392780/