I will be in Sicily for 10 days this July with my family which includes an 8 year old, a 4 year old and a newborn. I need to solidify my itinerary so I can confirm accommodations. It will be our first time to Sicily and we are super excited. Since we will be with our young children, we will care less about the history of Sicily and churches etc, but rather the food, the beautiful settings, getting lost in the streets and relaxing some by the sea. We fly into Palermo, out of Catania.

My original plan was to do 2 days in Palermo, 2 in Cefalu, 4 in Siracusa and 3 in Taormina. Now I'm second guessing my direction. Instead maybe 3 in Palermo, 3 in Cefalu, 2 in Ortigia, 3 in Taoromina. Anybody care to comment?

We won't have a car also.

If you are flying long haul, then having 3 days in Palermo could make sense. However as you have 2 young children and a newborn you might want to consider going straight to Cefalu and using that as your base for 5 or 6 days and then take the train to explore Palermo on a couple of days or if it’s too hot enjoy the beach.
Ortigia is really nice and there’s plenty to go and see in Syracuse. Do take a train trip to Noto. It is a lovely town and there’s a great ice-cream bar there close to the church.
Taormina has great views and good shopping. You can take a half day trip to Etna and there are some that are suitable for families with young children. You can also do a boat trip from there.
Enjoy planning your family holiday.

May be the sandy beach of Marina di Noto and the Roman villa (mosaics!) in the same area could get almost as much interesting as the ice cream bar.
Cefalu is a nice place with a good sandy beach. But it's more complicated to go from there to Catania:
taxi to Termini Imerese - train to Dittaino - railway company bus to Catania (all together about 4 hrs, whereas the direct Palermo - Catania bus takes 2 1/2 hrs)..

This July... you need to get a hack on as most places will be booked up already.

Palermo is really just a big hot city (yes it has some nice parts but I'd get out of town asap). Lack of car means Pulman buses (lots of them and very good) and trains. Ortigia prices will be high by now so Syracusa (general) may be more sensible