Ciao Cayala,
The easiest thing to do will be to arrange for the tours/activities yourselves with local hosts. Once you have booked a place to stay, many of whom actually will offer cooking classes and tastings directly at their place.
Many farmhouses that have a restaurant will offer cooking classes directly but some places have agreements with local chefs or restaurants for that service, so it is something you can arrange after.... or just take into account as you look at a place to stay at.
Keep in mind that once you're here, there are wineries practically everywhere. You can ask your hosts for local wineries where you can just call ahead and book a visit and tastings and go direct, without the need to join a tour.
As far as places to stay at, I'm going to recommend a few that meet some of your requests - either are a winery, which might provide food and tastings, or farmhouses/villas that do all (food, wine and cooking classes).
https://www.tuscanyaccommodation.com/romitorio-di-serelle.html - this one has a restaurant, offers tastings and has a restaurant on site. Now I don't recall but they might do the cooking classes as well, you should write and ask. They also have bikes you can use.
https://www.tuscanyaccommodation.com/villa-dianella.html - this is a winery resort that is perfect for food, wines and cooking classes. You don't mention ages of family members so here I am not sure if there are young kids, and in that case, I would not recommend it. It is a place perfect for adults, older kids, who will not get bored.
https://www.tuscanyaccommodation.com/podere-casanova.html - this one on the other hand is ideal for families with younger kids. Also provides cooking classes and tastings.
https://www.tuscanyaccommodation.com/la-topaia.html - provides food, cooking classes and bikes and it is in Mugello where there are many gravel roads perfect for biking away from the more traveled roads.
https://www.tuscanyaccommodation.com/fattoria-viticcio.html - this is another winery that would cover most of your needs. It is just outside Greve in Chianti city limits so it is perfect for also going out and go wine tasting at many other local wineries.
https://www.tuscanyaccommodation.com/agriturismo-fratelli-vagnoni.html - winery, perfect for kids, right outside of San Gimignano. So also ideal for going tasting in the immediate area, and there are many restaurants in SG.
Keep in mind most of Tuscany is hilly, so if you base yourself on a hill, cycling will be tough from the get go since it will be downhills and uphills right away. So consider how much biking you want to do when selecting a place to stay at. You didn't rate the importance of all the things you want to do, if you do, I can help you sort out which ones might fit better.
Hope this is a good part from where to start and help you out in your search. You have time, so let me know as you go along how your ideas change and I can help provide more info.