Well, from the City to Napa or Sonoma County is easily 1 hour 15 minutes to 1 hour 45 minutes (depending on location and traffic). It is a very congested route. So, it depends on how much time you want to spend on the road whether you stay there or in the city.. Also, if you are doing wine tasting, I strongly suggest that you either have a designated driver and/or have someone else drive you. Sampling at 3-4 places could easily have you end up pretty tipsy (or worse). Lots of accidents happen there for exactly this reason.
We did an airbnb in November that might work for your group -- it had 4 bedrooms (unusual in San Francisco), and two bathrooms and was very affordable, not to mention clean and had ample street parking available (very rare!) so renting a car would be possible. I'm happy to share specific details if you want. It was in the area of Golden Gate Park, so pretty convenient to get to the Golden Gate Bridge and points north.
The other option would be to spend your entire trip staying at a place in San Rafael, or Mill Valley. Lots of rentals available in both areas. And, an easy drive to the City (30 minutes or less outside of commute times), and would then substantially cut your travel time to Napa or Sonoma. There are also a couple of hotels in the area that I recommend to guests when they visit the area (I lived in the area for 20+ years). There are also a ton of wonderful things to see in Marin County (which is where these two towns are), including most famously Muir Woods and the Redwoods. The ideal way to see Muir Woods is FIRST THING IN THE MORNING, which would be easy to do if you stayed convenient to that. To get to the City, you'd drive OVER the Golden Gate Bridge....just like a local.
And, the Disney Museum is located right near the Golden Gate Bridge.
Location is very important "in" the City...some neighborhoods are walkable with lots of great dining; others not so much. Again, more than happy to steer you in the right direction.
Know that late June is sub-optimal weather "in" the City. Lots of fog. Tends to be on the very cool side. OTOH, in Mill Valley, it will be (on average) 10 degrees warmer and sunnier, and in San Rafael, 10 degrees warmer still and even sunnier. It's hard for people not from the area to fathom how different the weather can be in short distances, but it is true. My sister visited us once when we lived in Mill Valley, she complained that our town was "too cool" and then when we went to San Rafael, that it was "too hot." LOL. Same day. Napa can be downright hot in the summer months.
I hope this helps, and like I say, feel free to message me off line if you want more information or advice on specific rentals. I know Marin and the City very very well.
Also, plan to bring a hoody with you for the City. I've seen locals wearing down coats in the City in the summer...but that's overkill. LOL. But, I cannot stress enough the need for layers and long pants. Vendors on Fisherman's Wharf make a killing selling hoodies and sweatshirts to the unsuspecting tourists equipped with nothing but shorts and t-shirts.
