Don't know if this is the right place or not.

My cousin who is 16 wants to learn how to machine using a mill and a lathe. Are there any classes around the Bay Area for his age?

Can't really help in the bay area, but maybe look into something like Hacker Space? We have them out here in Sacramento and they give discounts to students. Not going to play with the big stuff, but they offer classes to teach the basics. It seems to be an evolution from computer guys, so maybe something like that in the bay.

Copy and paste from the one in Sacramento:

Maker Space
The Hacker Lab Maker Space is your community shop. Much like a gym membership, you can become a member, take classes to get trained up on the tools, then use the tools for your own projects. There’s 5,000 sq ft shop with plenty of workspace for you to hammer, glue, solder, weld and build. We even have a team of onsite makers to help you get acquainted and up ‘n running things. Featuring:
3D Printing Lab – featuring a Stratasys Dimension, and two PrintrBots
Laser Cutter – 80 watt 2’x3'
Wood shop – ShopBot CNC Table Router 4×8' + Saw Stop Table Saw and variety of other saws and hand tools
Metal shop – Mini lathe, grinders, cold saw, work benches
Welding – MIG, TIG, and Oxy welders
Electronics Lab – Oscilloscopes, power supplies, and soldering stations
Textile Lab – featuring two Juki industrial sewing machines

Try the CCOC in San Jose. If he is still in high school he can check with his counselor. They used to have night classes also. Been 30+ years since I was there, not sure what they have to offer these days.

I'd check into classes at the local Community College

Having taken 4 years of machining and machining related courses and subsequently working in for the last 11 years in a a machine shop; this would be my advice to anyone who wants to learn machining:

Apply to a shop, be up front about no experience, and offer to take a reduced pay while you train.

I learned more in my first month on the job than I did in my entire school term. There is also an expectation if you come out a school, a d some schools have developed bad names in the industry. NTMA is a good example, where the student's resumes all look the same (20+ years of experience in aerospace, CNC certified, etc) and they all flip because they've been taught mostly theory and we're never shown machining nuances that you get from experience. In fact, we don't even consider NTMA grads anymore because we always get the same thing.

Most general shops are gone now anyways, thanks to technology and overseas production - it's faster and cheaper to buy a new one than repair an old one. So most shops expect a certain level of on the job training for what they specialize in. Teaching the basics on top of that is a piece of cake. My entire schooling on the lathe was "This turns it on, this spins the chuck, this crank moves it this way, this one moves it the other." It's actually a good thing to come in with no experience because that's means there's no training scars to unteach.

So if he is serious about it, and he is capable to pick things like that up, just have him go in a ask at a shop. He'll get a better education than any school can provide.

Eta: sorry for odd grammar and speeling - fucking "smart" phones.

Chabot Community college has a really good machinist program. They have both a manual machine lab and a CNC machine lab. The guy running the program and I worked together nearly 40 years ago at LNLL.

Chabot is in Hayward. They also have a pretty good welding program and are restarting their vacuum technology program.

I live on the other side of the hill from Chabot. I've taken many of there classes.