Italy in 16 days
 

We are planning a September trip to Italy from the United States. Looking at going for 16 days, but figuring 2-3 days of that is travel time. Our initial interests are Venice, Rome, Amalfi Coast, Florence, and Lake Como. There will be two adult couples traveling with no children. We are open to staying in apartments, hotels, and/or rental houses. Trying to keep it budget friendly as much as possible, although we will pay more to get the cultural experience. Any suggestions are highly appreciated.
Also, any information on how many days you recommend at each place. We are interested in more of the scenery, nature than the city. Planning on only hitting the highlights in Rome.

Since you are interested in more scenery/nature than cities:

Fly into Venice: 2 nights (1 day)
Train to Varenna, on Lake Como 3 nights (2 days)
Train to Florence or since you like nature, why not base yourself close to a village in Tuscany like Pienza, Montepulciano (need a car) 4 nights (3 days). You can rent a villa or stay in an agriturismo.
Train to Naples. Ferry to to the Amalfi coast (choose Positano, Ravello, Praiano, Capri, other smaller towns etc...) 4 nights (3 days)
Train to Rome 2 nights (1 day to check out our highlights) fly out of Rome.

If you can find a cheap flight out of Naples, you can go from Tuscany to Rome to the Amalfi coast. But usually my flights are cheaper and with less connections from Rome. Also, I know you said only the highlights in Rome, but it is a very walkable city full of piazza and fountains. You might want to allocate more nights to Rome. But it is totally is your call.

Usually it takes 1 day over and 1 day back so you are left with 14 days. Florence - are you interested in Renaissance art and sculpture? If not, 1 full day to wander the city, see a couple of highlights might be enough, or skip altogether. Amalfi Coast - do you have an interest including a visit to Pompeii? With two weeks it will be very difficult to do all your mentioned destinations, even just seeing the highlights briefly.

A nice visit to all with minimums would be:
Venice 3 nights
Lago di Como 2 nights
Florence 2 -3 nights, if you want to include a day to the countryside/Siena
Rome 4 nights
Amalfi Coast 4 nights (it takes a little time to get there)

You may need to choose between Como and Amalfi. If you say you prefer scenery, maybe stay in the Tuscan countryside and just do a day trip to Florence, or skip it.

You also might not get fixated on Como. Maggiore is even more lovely, IME, less expensive, and easy to get to by public transport.

If Renaissance art and museums aren't your thing, I'd skip Florence and enjoy the Tuscan countryside.

Trains are best for all those places - well to get to say Amalfi or Lake Como - ferries once there are a great way to move around and book your long-distance train tickets ASAP for deep discounted fares - Acquista il biglietto con le nostre offerte - Trenitalia or www.italotreno.com - two competing rail systems using same tracks and stations - check both and book your own train on line- www.seat61.com has loads of helpful tips doing just that; general train info www.ricksteves.com and BETS-European Rail Experts.

I would consider dropping Lake Como and making the rest more leisurely.

First, realize you are underestimating the travel time. If you meant to say to visit these 5 destinations, your travel days would be closer to 6 days, not 3 days.
Moving around and rentals of most kinds are incompatible. Rental houses especially come with multiple day minimum requirements. Of course, you can chose to stay less, but you are on the hook to pay the minimum number of nights in which case. Also rentals come with stiff cancellation charges as well as restricted check-in time window.
Unless the scenery also includes architecture, your choice of destinations are not congruent except for Lake Como and the Amalfi Coast. Your target destinations are city heavy when your interest is "nature than the city."

The itinerary suggested by toujours voyager could be perfect for you. We have a trip report posted on our trip last fall to three northern lakes including Como. We have stayed multiple times in all of the places you are looking at, and we highly recommend airbnb accommodations. For 2 couples, you can get a nice 2 bedroom apartment, with a fully equipped kitchen, often for the price of one economy hotel room. In Rome last year, for example, where hotel rtes are high, we rented a nice 3-bedroom apartment for 7 persons for the price of one moderate hotel room. There are plenty of places that you can rent for even one night with flexible cancellation policies. Since you usually have to arrange to meet the owner, we use airbnb for stays of two or more nights and hotels for one night stays. Venice also can be pricey, so, the last time there, we went beyond airbnb and found a nice, relatively inexpensive canal front apartment on booking.com (which has hotels and apartments).