Taipei is very kid-friendly, locals are genuinely helpful and the majority of venues are equipped with baby-changing facilities, high chairs and the like. Taiwanese people are lovely and helpful, do bear in mind not much English is spoken. Ensure to have addresses in mandarin printed out or on your smartphone. A basic translation app will also help.
Taiwan has a tropical climate and it is best to visit during shoulder seasons. You may want to avoid typhoon season, which is September.