First trip to Italy must include Dubrovnik

Only way I know to do it is search. Skyscanner is a good site for sussing out European options. I use Kayak for multicity searches.
What parts if Italy are you interested in? There are ferries from Dubrovnik to Bari.

I just used Google flights to check a few random dates in October and was seeing 1 option for roundtrip Rome to Dubrovnik on Vueling for $90, though $135 was more commonly available on various dates. Some of the carriers want $300 and more.

Was pricing the challenge? Or timing of flights?

When I checked multi-city (SFO to Rome and Dubrovnik back to SF) there were a variety of options under $1000, most with 1 outbound stop and 2 stops on the return.

Rome is a must but depending on how long we decide to stay in Dubrovnik we would like to explore either Florence or Venice and/or make a long train trip at stay a day or two in Amalfi Coast or Cinque Terre, but it seems we would be pushing it. As far as flights, trying to find balance between price and timing. Would like to take red-eye at the beginning to arrive i the afternoon the next day ...

Have you considered making this a Croatia trip, and saving Italy for another trip?

Croatia and Italy is one of those country pairs, like Spain and Portugal, that seem "natural" to those of us in the US, but are surprisingly difficult to combine in a single trip due to poor transit links. You'll have to fly between the two countries, using much of a day; if you have to take a connecting flight, that's even more of the day.

October makes this even harder, as many of the flights to Dubrovnik are seasonal, and by October they won't all still be going. All you can do is play around with dates and cities on Skyscanner. Remember that flights may not be daily, and can be as infrequent as once a week.

I wish we could make this trip either/or, but we are willing to sacrifice time in Italy because my wife's wish is to spend time in Dubrovnik. One question about Skyscanner: Is it reliable booking trips through a third party not directly affiliated with airlines or a more known brand such as Kayak/Hipmunk/Expedia/Orbitz etc?

Use engines like sky scanner to get an idea of airlines and prices but I will always book directly on the airlines official website !

Flying is looking like your best option here.

If Venice was a place you'd like to see, the flights from Venice to Dubrovnik are a big cheaper than from Rome, but with a train from Rome to Venice that only makes sense if you want to spend some time in Venice as well. It's something I'd recommend in off season but would be hesitant to recommend in summer (but the same goes for Dubrovnik, really).

Excellent selection about going to and from Venice and working our way to Dubrovnik and back from there! We could take train to Rome and back ...

If you’re going to Dubrovnik, would you like this itinerary?

Fly from San Francisco to Split, Croatia (looks like Lufthansa has flight thru Frankfurt)
2 nights in Split
2 nights on Hvar Island
2 nights in Dubrovnik
Fly to Rome
3 nights in Rome
Train to Venice; 2-3 nights in Venice
Fly home

Or you could fly from Dubrovnik to Venice and end in Rome since most flights are early from Venice, and you need to go from the islands to the airport on the mainland.

This would give you some nice locations on the water since you’re in the area.

My last trip to Dubrovnik was through Rome on Delta and Alitalia. I suspect you will save money going US, Dubrovnik, Rome, US. For my taste 3 nights in Dubrovnik was enough. That leaves 6 for Rome and 3 for Florence more or less.

I love Croatia and can't wait to go back but I agree that it might be best to save that trip for another time

As for booking flights thru a third party - keep in mind that airlines do not allow anyone to discount their published fares. Any flights you find via an online TA, aggregater site, etc, you will find the exact same fare on the airline's website. As to the legitimacy of Skyscanner, it is a reliable search/comparison site.

I like James E's itinerary. Every time you change locations, you lose half a day of sightseeing time. It will definitely save some time if you fly into Dubrovnik and out of Rome. If you really want to do a day trip, you could do a very long(easiest on a bus tour) day trip from Rome to Pompeii and Amalfi coast .