If you intend to stay right along the Adriatic coast, originating in Venice or Bologna, train or car are equally good choices. The autostrada runs near the coast, with many sea glimpses and beautiful views. The road (a toll road) is well maintained and well signed, and very easy to drive. The train line runs even closer to the coast and there is a medium-fast train (Frecciabianca) along the entire route.
If you decide to head east from Rome, forget the train. There are no Freccia lines, it's all regional with multiple changes. Again, the autostrada is easy to drive on with spectacular mountains all around. I've been on it 3 times in the last week and while there is a lot of summer construction, it is not congested. You could take a comfortable coach bus from Rome to Pescara or Teramo if you want. Try Prontobus, it's one of 3 or 4 coach lines that do the run. They actually make fresh espresso shots on the bus and hand them out to the passengers en route!
However, if you want to explore the interior at all, such as Civitella del Tronto, you will need a car. The public transit options are really limited to the coast. There is a regional coach bus service that connects all the small towns, but it's meant for school and work trips and is not as frequent as you might want (check the TUA website.) You can mix and match by taking the train or coach bus to Pescara and renting a car there. The train station has outlets for all the major car companies.
I'm biased, as I love the mountains, but missing the national parks and historic villages away from the coast would be a shame in my opinion. The smaller state roads (SSxxx) are also surprisingly well maintained, although only one lane each direction and limited shoulders. Once you get off the SS roads and onto purely local roads, it's a little more rough but is usually just for very short distances.
If you decide to come into the interior, I'd be happy to provide some sights for your consideration. But Civitella del Tronto is as beautiful as any, and as you have family history there, you might be better off to just settle in and not roam too far for that segment.
Driving in Abruzzo is nothing to worry about. It's far, far easier than driving in the northern lakes area with its congestion and bad roads. And the sights around each bend are breathtaking. Have a wonderful trip.
