Question - What is the best way to fly from NJ to Dubrovnik?

As a starting point I'd go to Google Flights, enter whichever airport is your preferred starting point (Newark or Philadelphia, I guess?) and Dubrovnik as the destination. If the results look ugly, you can try Zagreb.

Do you also plan to fly back home from Dubrovnik? With most itineraries, it's useful to fly in and out of different cities. To do that, you would select "Multi-city" rather than "Round trip".

The best way is first class; otherwise, like acraven, I use Google Flights to discover the options.

Zagreb than Dubrovnik by Croatia airlines. You need to check who did contract with DBC and from where they are flying.

According to your other thread about Dubrovnik, you have a travel agent working on a plan. Why not ask the TA to do the flight research, too?

My trip I went through Rome I think

I was in search of a Raleigh to Dubrovnik and couldn't find any reasonable prices so we opted to go from Ral via JFK to Paris then to Dubrovnik. Total cost was $550 ($350 to Paris then $200 to Dubrovnik). We decided to stay in Paris a night for fun. There is an luggage storage area at the airport. It may not be worth the hassle of an airline transfer but it was a lot more affordable. We found our flights on skyscanner by breaking them down by flight legs.

We will be flying out of Ljubljana, Slovenia which looks to be about $700 one way.