Italy was our favorite country pre-child, and we learned over 4 trips the following:
. Enjoy getting there and enjoy wandering back. First couple of trips, we had a packed day scheduled each day. By the 3/4th trips, we just said: We'll meander over to x, slowly enjoy and meander back. Everything is so lovely and has so much meaning that rushing through it to get to the next monument means you miss 3/4s of the culture and the beauty.
. Italians enjoy the whole eating experience. Don't expect fast service and a waiter who replenishes your glass every nanosecond. Going out to dinner is a full night in and of itself.
. The evenings/nights are wonderful. Walking around in the half-light, seeing all the tourists, seeing many of the Italians who just love style, is just a wonderful way to see any city.
. Take trains from different points. Just looking out the windows as you travel is a great experience. There is nothing like the Italian countryside.
We never had any issues with theft on any of our European trips. I bought a Baggalini "mini bag" (I think that is the way your spell Baggalini) and carried it flap side inward inside my coat. Don't take your whole wallet/passport etc., with you...I wouldn't do that in Italy, any more than I would do that in NYC, Paris or London.
In short, don't go the 100 mph that Americans seem to feel they need to so they don't "miss" anything. Enjoy each day like an Italian..which is leisurely and savoring the moments.