Definitely head into SLO. Wineries down there are very family/children friendly for the most part. It's not as slammed at Napa. My wife and I wine taste quite a bit with our now 2.5 year old but probably go to a winery 1-2 times a month since he was a few months old. My wife and I also went to Cal Poly SLO for school.

Best bet is to not try to taste with the kids. Just buy a bottle of wine to drink outside, bring some snacks, and go sit outside. Or what we've done is to have 1-2 people do a quick/shortened tasting to decide on what wine to get. I'd look for wineries that have good out door space. Bring a blanket to put down on the grass and a few toys (avoid brining balls, they end up being trouble).

Edna Valley in SLO is one of my favorites, and there are a few other wineries right there like Baileyana and Tolosa. Tolosa has a large outdoor grassy area you can hang out at (but smaller/modern tasting room), and they also sell food. One of my favorite wines in SLO area is Laetitia but I've never actually visited. Had friends that visited and have drank lots of their wine though.

I wouldn't plan on going to more than 2 with the kids, and you could also make a trip down to Pismo Beach for some ice cream or clam chowder at Splash Cafe (best in the world).

I've only tasted in Paso Once or twice, and don't remember much about the wineries. There are some good ones though. Paso is typically 15+ degrees hotter than it is in SLO, so you may be hitting close to 100 over the weekend, while it's mid 80's in SLO and mid 70's out in Pismo. Huge temperature change once you go over the hill and then down to the ocean.