Kathy's advice is excellent. Until you go there, you cannot fully understand just how dense Italy is. It's tiring, in a good way. But there's a lot to take in everywhere, and rushing through seeing lots of places means you don't enjoy them at all.
I like her idea of seeing either the north or the south, but not both, on your first trip. My first trip to Italy (of seven, and counting) was to Milan, Venice, and Florence, with day trips to Siena, Lucca, and Pisa. Many were scandalized - "how can you go to Italy and not see Rome?" But I wasn't interested in Rome at the time. However, the first trip kindled that interest, and I made it the focus of my second Italy trip. Now Rome is one of my favorite places, but who knows how I'd feel if I only went there out of obligation?
No matter what else you do, do NOT just have one, jet lagged, arrival night in Venice. Either spend more time there, or save it for another trip.
As for the airports, the problem with small ones is that (coming from the US) you can only reach them by changing in larger ones. I personally find changing in large airports harder than arriving or departing, but of course it does depend on the airport and not everyone agrees on this.
I don't know where in the US you're departing from. Venice has a very few nonstop flights at certain times of the year (Newark, Philadelphia, Chicago, JFK, Atlanta). Rome and Milan get nonstops year-round, but even from many large cities, you will have to change. If you do have to change to get to Italy, most prefer, if at all possible, to change in Europe. This is not only faster overall, but gives you more options. From, say, San Francisco to Venice: if you go via Newark, there's only one flight from Newark to Venice a day. Miss that for any reason, and you have to wait a whole day. But if you go via Frankfurt, there will be several Frankfurt to Venice flights. As long as you're flying all on one ticket the whole way, the airline will put you on the next available flight at no charge.
If you say where you're flying from, others can advise you on the best options; it really is different for each origin city.