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Reykjavik, by far and large, is the cheapest city to fly to Europe from the US. The problem, however, is that Iceland is far from continental Europe so you would need an additional flight to go anywhere else.
Use google flights, put your airport and leave the destination blank or just put “Europe”. You should see a map of all the places you can fly to with their prices. Usually Madrid and London are cheap to fly to but I’ve seen Lisboa and Rome as well.
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From Rome just take a train to Naples. For Dubrovnik take a low cost carrier. Flights into Italy are generally cheapest to Rome and Milan. To get to Amalfi it's just a quick flight to Naples or a long train day. To reach Croatia, you could fly to Zagreb then train to Dubrovnik.
AMS (Amsterdam Schipol) is one of the cheapest to fly into; it's a major hub so if you want to go somewhere from there it's not too hard. If you want to save money, avoid London like the plague. Amsterdam can be cheap if done right; believe it or not, staying in the red light district isn't bad and is a bit cheaper than going to the major hotels.
Reykjavík and London are consistently cheap cities to fly into from the US. Different airlines have different pricing models. Discount airlines can have their lowest airfare as far as 6 months out.
Use Google Flights to explore cheap flights to anywhere. Search destinations and track prices to find and book your next flight.
