My 5 year old loves going to all events, even just markets and any that have music. She doesn't like to walk too much but this way gets distracted and ends up doing a fair amount of walking during the day.
1. Just over 30 minutes away (north-west) is Montespertoli. From Saturday 25 August to Sunday 2 September there is the Sagra della Pappardella - Pasta with hare meat sauce as well as many other sauces. Dancing on many evenings.
2. 1 hour and 30 minutes south passing by Siena and Montalcino (stops in themselves) you arrive at Bagni San Filippo where there are free natural hot springs which will probably amaze your kids.
3. Saturday 1 and Sunday 2 September - the Antiques Fair at Arezzo - one of the biggest and best known in Italy. Maybe not for the kids but Arezzo itself is very enjoyable.
4. Florence - Piazza del Carmine hosts the Florence Street Food Experience. Must see – the frescoes of Masaccio in the church of Santa Maria del Carmine
If you are finding that the days are just a little too hot - you might want to check out a visit to Vallombrosa, located in a wonderfully refreshing ancient forest - great for a picnic, running around and exploring nature, while giving you an opportunity to visit a place of historic importance.
I took my nieces and nephews there one summer (2 - 4 - 6) and it was something they really enjoyed without overloading them on Italy.