Abruzzo Coast

Sorry to say, it does appear that the train from Rome will be your best option. Flights into Pescara seem to be primarily from alternate airports in Europe (i.e., Frankfurt Hahn rather than Frankfurt am Main, London Stansted rather than Heathrow or Gatwick). There is one flight a day from Malpensa (Milan) and two a day from Milan Linate (which means going from Malpensa into town - like in Rome.

I would rent a car. That way you can explore the many sea towns and mountain towns and one of Europe’s most beautiful national parks.
https://abruzzoturismo.it/en/
http://www.italia.it/en/travel-ideas/the-mountains/the-national-park-of-abruzzo.html

I'm headed to Abruzzo in 3 weeks, flying into Rome. I've been advised by people who live in Abruzzo to avoid the train, as it takes forever. They suggested taking a bus right from the FCO airport to Pescara. Options are Prontobus, DiCarlo Bus, and Flixbus. It's supposed to be faster, cheaper and comfortable. I'll find out soon!

I don't know where you're located, but if it's easy/cheaper to fly to Dublin, RyanAir is the airline that flies to Pescara most often. They are of course based out of Dublin. Don't know if that's feasible for you.

It will be my first time in Abruzzo, so I'm not qualified to comment on beach towns, but my acquaintances live in San Vito Chietino and they say the beaches are great.

It looks like one can pick up a Pronto bus outside baggage claim. It’s 3 1/2hr ride to Pescara in an air-conditioned coach bus. I need to figure out if we need to reserve beforehand or we just hop on and pay? They leave every hour or so. https://booking.prontobusitalia.it/public/ricerca.jsf?partenza=12&arrivo=1&data=20082019

By the time we get to Pescara, after having flown from the US the same day, we may end up staying in Pescara a couple of days. We probably will need to rest before relocating to a smaller beach town.

I don't have first-hand knowledge of how busy the buses are. I intend to make a reservation just so I don't have to fumble around with money after getting out of the airport. But unless you're going in August, my guess is on-the-spot ticket purchase is likely fine.

One other comment - the only useful train line in Abruzzo, I'm told, runs right up and down the coast, so you can get on/off at most any beach town. As long as you're not trying to get east-west, the train should be fine once you get to Pescara.

And finally - the only guidebook I've found that covers Abruzzo in detail is a Bradt guide called Italy: Abruzzo. I've found it very good for my trip planning so far. It discusses the various beach towns in detail.

I realize you asked about Pescara, but I just want to share that flights to Ancona from Dulles on Lufthansa can be quite convenient (one painless plane change in Germany, was affordable as a direct flight to Rome in my experience). The convenience level might be higher as well, as there are numerous attractive small towns accessible from Ancona by train as well as bus. You might take a look at Sirolo.