I just wanted to let you that I have been traveling with my children since they were 4 month old and that Venice and the Amalfi Coast were very easy when they were 15 months. I wanted you to know that it can be done easily if you are in good shape and you are able to carry your baby in the carrier for 1-2 miles every day.
The nice thing about Venice is that it is car free. So in the areas of Venice away from the crowds, we could let her walk, run after the pigeons, play with the water fountains. In the Amalfi Coast, Ravello is pretty flat, again car free, and Positano and Capri both have a pedestrian center.
While hiking in Cinque Terre with my 2 years old, that is where she started her love affair with octopus salads when we stoped for lunch in Vernazza.
Make sure you have a baby carrier (like an ergo carrier), a very small stroller (like pockit that can be brought in on the plane in the overhead compartment) -ours could sleep in the stroller while we were eating dinner.
Many airlines have a baby bassinet that you can use free of charge. It is amazing because you don’t have to hold her in your arms the whole flight, and you will be able to get some sleep. There are weight and height restrictions. Make sure you call the airline so you can reserve the seats that have the capacity to have the baby bassinets. They fill up quickly, first come, first serve.
If carrying the baby is too hard for your stamina or if you have back problems, then yes I agree save the Amalfi Coast or Cinque Terre until you child turns 3.
What ever you decide, I agree that with 8 nights, do 2-3 destinations at most. Also try to see if you can do one carry-on and a medium backpack for both you and the baby. Baby clothes are so small, and diapers can be bought in Italy (just bring enough for the first 4 days).