I am traveling to Rome this July and am looking for best options for getting from the airport (Fiumiciano) to my hotel on Via Auerlia, approx 19mi apart. I understand taxis are very expensive. Can anyone give suggestions?

We've been to Rome several times and have always used limoservicerome.com
I believe others on this board recommend them also.

A taxi will cost about 55 Euro. I just got back and we took a private car that was very efficient for the same amount booked through Icon Tours. If that's too much (and it likely is with the Euro being so strong!) there are city buses that take you into the city center which will be cheaper. A taxi within the city center is relatively cheap (8-10 Euro).

Via Auerlia or Via Aurelia?

I can't find a Via Auerlia.

A taxi or arranged private car service would cost 40-50 euros depending on the service. For two or more people, this cost might come close to the other options I'm about to list.

There is a shuttle express train that goes from Fiumicino to Roma Termini the main train station, and costs about 11 euros per person. But then you must get to the hotel.

If on Via Aurelia, it is across town from Termini, so you might take a taxi and spend 10-20 euros getting there.

If on Via Aurelia, Roma San Pietro station is not far away, nor is Roma Trastevere station. There may be local train connection from the airport to one of those stations, and then you'd have a shorter taxi than from Termini. The local train might cost half the price of the shuttle express.

Have you asked your hotel what they suggest?

It depends on what you consider expensive and how much trouble/time you are willing to take. The limo services seem to cost about 35 euros.
You can get from the airport to central Rome quite easily for 6 or 7 euros with a combination local train to Trastevere and tram. Getting to Via Aurelia would involve changing to another bus line and spending another euro and more time.
Note: I think you mean Aurelia, not Auerlia--near the Vatican?