In 2019, the first direct flight between the United States and Croatia finally took off from Philadelphia to Dubrovnik via American Airlines, although it's since been grounded due to COVID-19 and its future is still undetermined. Usually, travelers from the U.S. must fly into a European hub first—the best options are London, Munich, Frankfurt, Madrid, or Vienna—and then transfer to a flight to Croatia. Croatia Airlines has many affordable options, or you can check budget airlines such as EasyJet, Ryanair, or WizzAir; just keep in mind that they have stringent baggage limits, offer few free in-flight services, and you'll likely have to reclaim and recheck your bags at the airport, so make sure to leave plenty of time between flights.
Italy is another great choice for a stopover because there are many direct flights, buses, and ferries connecting to Croatia. If you're traveling to Istria or Kvarner, consider a flight to nearby Venice (2½ hours to Rijeka by car) or Trieste (1½ hours by car), and then arranging a car rental or bus. Or if you're going to Dalmatia, you can take an overnight ferry across the Adriatic from either Bari or Ancona.
Within Croatia, flights are very affordable in the low season. You can travel between Dubrovnik and Zagreb, for example, for under $100, which is a comparable price to taking the bus but much quicker. In the high season, it's worth looking at domestic flight options if you want to save some time and cover more ground, although buses or a car rental will likely be cheaper then.
There are airports in Dubrovnik (DBV), Osijek (OSI), Pula (PUY), Rijeka (RJK), Split (SPU), Zadar (ZAD), and Zagreb (ZAG). You can fly direct to all of them year-round from international destinations, although in winter most international flights go to Zagreb.
Ground transportation to major cities from airports in Croatia is well organized and user-friendly; shuttle buses and taxis operate at all airports year-round. Tickets can be purchased online, at the airport, in the city center or from the bus driver.