It depends if you are looking more for cities, small towns, castles, nature, beach, museums, activities (hiking, sports...) or even food, but in all cases Italy and France are certainly the most interesting countries to visit.
In Italy, don't miss Rome, Florence & Pisa, Venice and Naples & Pompei. In France, Paris is obvious, otherwise, if you go north, Lille, Mount St Michael, west, the Loire Valley (best Renaissance castles in Europe), or south, Nice, the French Riviera, Marseilles, Avignon, Nime, Carcassone...
For landscape and architecture, Spain is very nice as well. Don't miss Salamanca, Madrid, Sevilla, Granada, Cordoba and maybe also Merida and Caceres (all center to south-west). Just beware that both Italy and Spain can get very hot in summer (usually 30 to 40 degree Celsius, i.e. 85 to 105 degree Fahrenheit).
If you don't like the heat so much and want to see a lot of history without spending too much money and time in public transports (or car), I recommend Belgium. I have just written a
new guide to Belgium and revisited a dozens places this last 2 weeks. Belgium has probably more gothic cities and medieval castles per square meter than any other country in the world. And food is as good as in France, although probably easier to adapt to for Americans.