Rome to Amalfi Coast

Quickest way would by car but I wouldn't advise a 3 hour drive after flying in from LA. Here's a link to Rome to Rio Website that shows some options.

When you say it's "most cost effective to fly into Rome as opposed to Naples," that may be true if you're just looking at flight costs. But there are other costs to consider.

1) Getting from Rome to Sorrento or to the Amalfi Coast is not free. You have to add some combination of train fares, boat fares, bus fares, or taxi fares. If you are using taxis, the cost adds up quickly.

2) The time, hassle and aggravation must also be considered. There's no "one seat ride" from the airport (other than a pre-booked taxi which will cost several hundred euros), and as you say you will be jet lagged and tired. Rome's airport and Sorrento simply don't have easy or fast connections, and the Amalfi Coast is a further connection from Sorrento.

I'm not saying you shouldn't fly to Rome instead of Naples; I know that it can sometimes save a huge amount to use different airports. I was just looking for May 2017, and for some reason, nonstops from New York to Geneva cost a fortune, while nonstops to Zurich - on the same dates - are not expensive at all. I'm also aware that while there's a nonstop from LAX to Rome, there are no nonstops from LAX to Naples, so you'll have to change at least once.

But if it's not much more to fly to Naples, do it. From Naples airport, there's a bus that goes right to Sorrento - the "one seat ride" I was talking about.

This question has been asked and answered numerous times.
I don't have specific times but know there is no great way

Ranking in terms of speed
1.) private car service (most expensive)
2.) rental car (not a great idea for this route, complications on both ends)
3.) train ; you cannot take a direct train so need to figure out if routing through Salerno and then working back via bus or ferry is better for you or routing through Sorrento which still requires a change in Naples is better for you. From Sorrento still need to catch a bus unless staying there.
My guess is if staying in the town of Amalfi, Salerno is better. If staying in Sorrento or Capri. it would obviously not make sense to go to Salerno. If staying in Positano or nearby it is probably a wash and becomes schedule dependent.
4.) Bus to Sorrento (often requires a taxi to get to the bus station in Rome since it is a little outside the tourist areas, which although adds costs can be a good thing)

We choose the bus for a few reasons (mainly because I thought it was easier and less stressful than the train options) but I have no doubt it was much slower than any other option. We did not leave from the airport though.

If I were attempting to do what you are (fly from California and upon arrival make it all the way to Amalfi Coast same day) I think I would opt for what must be a super expensive Private Car Ride ; the other options would just mean a nightmare travel day.
Either stay a night in Rome and do 2,3 or 4 the following day or don't stay a night in Rome and go with #1 ; just be aware the cost difference will I imagine be more than the cost of a hotel for the night in Rome.

If you can fly into Naples, it is much easier from there to do same day of arrival.
Nothing from the US is flying directly into Naples so it does likely depend on your connections and schedule options.

Time IS money--have you actually checked flights into Naples? From the east coast, I have often found flights to Naples with one brief layover cheaper than Rome flights. Check Lufthansa.

There's a new bus service from FCO to Naples. That might help by eliminating at least one transfer (FCO to Termini).

Are you specifically going to Amalfi (the town)? Private transfer from Naples would be the fastest whether you fly into Naples or Rome. Jay (from Chicago) recently posted about that to get to Sorrento.

Thank you everyone for the responses! I am looking for flights into Naples, yes and will compare all costs to make the best decision. The bus from FCO to Naples looks promising and then from there we can get a private car from there. We will probably be in the city of Amalfi the first night, yes. Unless someone has some suggestions (thinking 2 nights in Amalfi/Positano and 2 on Capri)