My favorite neighborhood is hands down Sempione. It's tucked away behind lush Sempione Park and is filled with charming cafés and tiny shops. It's less hectic than the center (Duomo) area and there are plenty of things to do with kids.
Starting off at the metro stop Carioli, head towards Sforzesco Castle where you can admire the fountain, the regal courtyards and the imposing towers. The castle is home to a number of interesting museums and libraries too such as the children's museum Sforzinda where you can find tours and workshops for kids aged 4-13.
Right on the edge of Sempione Park next to the Civic Arena is the Civic Aquarium of Milan. It's housed in a beautiful liberty style building and is the third oldest aquarium in Europe. Here you can see over 100 varieties of fish and the vegetation of Italy's lakes, rivers and coasts.
Inside of Sempione Park near the Aquarium is a playground with space to ride a bike or roller skate and rides. There's a train and bumper cars too.
Another good spot for kids is the Triennale Design Museum. Here you can find art exhibitions and creative workshops for kids. You could head down to the fenced in garden too where kids can run around and play (but can't escape) while you sip your well-deserved spritz.
If you're in the mood for a quick bite, in the middle of the park is Bar Bianco - an outdoor café that offers coffee, sandwiches, salads and drinks. Otherwise if you keep walking towards the Arco della Pace (the Arch of Peace) in the pedestrian area of Corso Sempione there are tons of kid friendly restaurants such as Jazz Cafè, Living, Duomo some of which offer entertainment for kids during Sunday brunch. The best gelato can be found in Via Cagnola at Gusto 7...they use only seasonal products and come up with some unusual flavors too such as basil, pumpkin or ricotta.
Finally, let's talk shopping. Do you know what else is the Sempione area? Apple Pie - the best baby boutique in Milan is right in via Cirillo. Here you can find unique gifts and toys, clothing, accessories for babies and children. Stop by just to say hello or just to tell us how good your ice from Gusto 7 was ;)