Flying to Croatia from Los Angeles

Head over to one of the major travel sites like kayak.com, or expedia.com they give you all the possible connections available. For open jaw select "muli city" instead of one-way/roundtrip.

Croatia Airlines is part of the United/Lufthansa group. If you fly on one of those to Frankfurt, you can take them from there to Zagreb (sure of that) or Dubrovnik (I think I'm remembering correctly.),on the same ticket. Start at one end of the country and end at the other. In May, I'd start in the south.

Definately go through Frankfurt, much better airport then Heathrow. There are only two major airports, either Zagreb or Dubrovnik and you have to go through Zagreb to get to Dubrovnik. the previous poster mentioned to go south first, they are correct because it does get extremely hot in the summer. It is not easy getting around Croatia, but better for driving since they have done major improvements to the roads and built new ones. Every time we visit we use different combinations, bus, train, ferry, or private driver. the way we go is first we decide where we want to visit, when, then figure out the most efficient way to go (it doesn't always mean the cheapest). Sometimes the train is a longer trip but the times or drop off point might be better. If we are not in a rush, we have taken buses but if we have a time issue, we hire private drivers, especially when we are 6-7 people. You will definitely enjoy the country, the beaches (mostly rocks or pebbles), the food, and the people!

Yes, use a search engine like kayak or skyscanner. The best and cheapest routings are not always the obvious ones.

There are more than two airports. Split is about the equal of Dubrovnik, and because it serves a population about five times that of Dubrovnik, tends to have a wider range of routes in the non-peak season.

We traveled to Croatia in May of last year. I agree that starting in the south is good from a weather perspective but I found it logistically too difficult to do. Croatia's high season is July and August and a lot of their transportation is limited during May. Buses and ferries only run on certain days. We did a combination of buses and car. The bus we used to get into Croatia from Trieste. We took that bus directly into Rijeka where we rented our car. We then took the bus again from Rijeka to Rovinj and then a ferry over to Venice.
A car is the best way to go as someone else has mentioned. The new highway was wonderful and provided some lovely views.
Best of luck!