Hello gjturley, and welcome to the forum!
My advice is learn what you can and then use it. Don't let the hesitancy stop you from saying buongiorno or buonasera when entering a shop or business. First it's a courtesy and good manners and it will also let them know you don't speak Italian from your pronunciation so they can adjust. Italians are happy you're trying and will work to understand you as much as they can. I've been slowly learning more Italian and it is a complex language and I guarantee you will get things wrong but Italians won't generally correct you (unless you ask) and if they can understand what you're saying they'll play along so you get to practice.
The tourists that come into a shop, say nothing, touch a few things and leave without saying something are looked down on. But those that come say 'Hello' when they enter the shop, acknowledge the people that work there and then say 'Goodbye' when they leave are considered good customers even if you don't buy anything.
But the Italians are talkers so the thing that comes up again and again is don't let the fear of saying the wrong thing keep you from saying something. As I heard an English girl who was in Rome and spoke no Italian said "If you say nothing they assume you are bored or boring. Just say something so they understand you're trying to engage. Personally this is a hard lessons for me so that's why I have paid so much attention to the topic.
That said I also have to second they concern that your schedule seems very fast paced. Keep in mind 2 nights in a location is only really one full day. The rest of the days you'll probably have your bags with you and you'll be spending time travelling and checking into/out of hotels etc. With only one full day in the big 3 cities of Venice, Florence and Rome this seems like just a quick sample of places that could really benefit from more time.
Have a great trip,
=Tod