What other Countries while visiting Italy?

It is expensive and time consuming to try to jam long distances into such a trip. two weeks is a good amount of time for the highlights of Italy. On our first trip to Italy 35 years ago we did Venice, Lago Maggiore (Stresa), the Cinque Terre, Florence and a week in Tuscany. We had 3 weeks and didn't even manage Rome on that trip. If you want to do another country, I would recommend flying into Paris and out of Rome flying in between from Paris to Venice or Florence. But with so little time, it would be best IMHO to stick with Italy.

With all due respect, a map says it all. The borders are along the north. Beaches, mountains, seaports, strudel, lakes. They are all good countries. Do you want history, museums, hiking, sitting around looking at scenery? I'd recommend figuring out what you want to see/do first then pick a country. If even. As has been posted, you could spend two weeks just in Italy and see enough variety as you would in six different countries!

You can do it, preferably with an "open-jaw" itinerary -- i.e., from the US to one European city, then return home from another, to avoid backtracking.

For example, my grandson and I just returned from a ten-day trip. He's a student of Latin, and is interested in both ancient Roman history and WW2. So we flew into Rome and spent four nights there; then took a train through the Alps to Munich and spent five nights there, making daytrips to various parts of Bavaria. Then we flew home from Munich.

OK, so technically we did see two other countries, Austria and Germany, but we never got off the train in Austria. The trip was spectacular, though.

I think "2 more countries" is a little ambitious given the 14-16 day time frame you mention.
Italy is worth devoting 12 days to as a minimum - Venice(2) Florence(3) Amalfi(3) Rome(4).
Two ideas that came to me ...
1 Austria - 2 nights each in Vienna and Salzburg
2 Switzerland - Geneva(1) => Zermatt(1) => Glacier Express Train to St Moritz(1) => Zurich(1)
http://www.glacierexpress.ch/en/Pages/default.aspx

How about Vatican City and San Marino?

Larry - A classic subtlety!!!

Vatican city comes 'for free' if you are visiting Rome. San Marino is not convenient for any of the big names to visit in Italy, but it is convenient for the beaches of the Adriatic.
To add to djp_syd's comments. Vienna and Salzburg are a long way north of the mains tourist places in Italy (Venice, Florence) I don't think you have time in 14 days.
Don't bother with Geneva or Zürich. The other suggestions are good (Zermatt => Glacier Express to St Moritz), but I would never do less that 2 nights in one place, and only if you already have north-west Italy in your plan.

One can easily spend 14-16 days in Italy itself. Splitting your time between Venice, Florence, and Rome would be our suggestion.

You don't say where in Italy you will be. Without that info, it would be difficult to recommend any other country(ies). Have you traveled to other European countries before? And, when will you be traveling? If you are in the north of Italy, I think it can be done more easily than in the south. What is your mode of transport? Are you thinking of trains or planes to get to the other countries?

Larry beat me to it.