Restaurant San Mateo to Palo Alto.
I too endorse Left Bank Menlo Park. We always enjoy their food (seasonal fish, lamb burger & steak frites are usual favorites) and friendly service.
Also, I was just at Costco & was reminded of some discounted gift cards they sell ($79.99 for $100)...
Max's Redwood City/Palo Alto
California Pizza Kitchen
Seeing that you're a resident of S.Ca I'd recommend Max's. They are a N.Ca chain that fits the description of what you're seeking. There are several locations to choose from, 3 in the vicinity you mentioned and the menu is vast to cover the food taste of all ages.
I'm confused because a bistro doesn't have a particularly large menu.
I just typed "san francisco bistro" and got L'Adroise, which has a very nice classic bistro menu, and bistro ambiance.
No burger. And not so good for 12 year olds, or 72 year olds who might have trouble hearing.
So --- are you really looking for classic american style cooking with a very broad menu ( like a diner, which is where the Max's rec comes from, but Max's RWC is pretty low end )?
On the south end of your scale, my go-to is Zibbibo. Benefits are it's not that expensive, it's HUGE so getting a table for 10 will not be a problem, the food's very generally decent, and it's not loud.
In Menlo Park I would send you to Left Bank. It's actually a bistro, but with a broader than average menu, is also quiet, and has very good drinks (which you didn't mention, but most times if there are 10 people and a family gathering, at least someone will be happy with a drink).
In RWC I would probably have to go Donato Enoteca. I'm not as wild about the food as some here, but the food is italian and a step above. Reasonably quiet. Harder to book a large table.
In San Carlos, I have a soft spot for Piacere, but mostly because it has a lively bar scene. Town is more packed and louder and more adult - unless they have a back room, I wouldn't consider it. Mythos is more middle of the road, quieter. If you're in for a thrill consider Red Hot Chilli Pepper, which always has a lot of kids and larger families, because indian families go there when the parents are homesick for desi chinese.
Paul Martin's is a new & expanding chain. I've seen in the one in MV but haven't been in. I think it's purpose built for exactly what you intend, but I've never been.
San Mateo - this town is so chock full of chinese and japanese that unless you want a huge table at Little Sheep it's hard to make a recommendation. Central Park Bistro is tolerable but I've only had one meal and it was a while ago. Kingfish has about the right atmosphere but the menu's very long on fish and short on everything else. There are a few other italian places I haven't been to and haven't seen reported - for people like me who live a few towns away, the only reason to drive to san mateo to eat is for its first-rank japanese and very solid chinese.
One other hidden gem. The sofitel restaurant in Redwood Shores is really pleasant and easy to park at. I've only had breakfast and lunch there, but if you're looking for a soothing atmosphere, you'll do well there.
And, finally, you should strongly consider Buck's in Woodside. Unless you really want to stay near 101, Buck's has it all. Some big tables in the back, corny american with a real silicon valley twist ( Buck's is a real "scene" place where companies get pitched during breakfast meetings ), and actually good food in a relaxed environment. The online menu doesn't look that huge but I always am happy with whatever I order there (today I would order a crab dish - we're right at the business end of the local crab season). To get a feel for the place, type "buck's of woodside" into Flickr, more than going to the website.