
Shopping with small children is not easy! But can be okay if you give them some playtime in between.
Of course if you choose to shop with and for them, that's a different story. Toy shops are not that common in Taiwan, but the biggest, "Toy's r Us" has 16 outlets around the island so you are likely to find one close to where you are.
IKEA department store is another outlet that is good for children. They have 4 stores in Taiwan, three in Taipei area and one in Kaohsiung. They carry a lot of items for children and some toys. When you are ready for lunch they have a special children's menu and high chairs in their restaurants. What's more is that they have a special "day care center" where your child can play while you do your shopping.

ESLITE bookstores in Dun Hua and Xinyi as well as Tianmu districts have a children's section and a lot of "happenings" in the shop that can be of interest to small children as well. They also have a café, gift items, toys, a paper section and more.
Department stores like Sogo, Mitsukoshi, Far Eastern, New York New York among others have nice children's clothes. Special shops for the little ones are also available in Taipei and carry some international brand names.
In a way, any night market is like a "crazy" area for children, but normally open after 5.00pm and open late at night. Simple games, small cheap toys and children's clothes are available. Be aware that there are an awful lot of people and it can sometimes get overly crowded.
Ximending, an area close to Taipei Main Station, is another place your little one might think is okay. (Mind you, small children never like to shop.) Here are massive outlets for everything! Mostly for young people, and with that comes the unusual.