If you don't have a car, getting to Vasto is quite easy!
You will need to catch a train from Roma Tiburtina to Pescara Centrale and from there, take another train to Vasto.
The first train leaves at 7:45, then 10:36, 12:46; 14:03 and the last one you can take without arriving too late is 16:12 because the trip is 5-5h30. It will cost EUR 16,94. I must say it is quite worth it because the train is crossing the scenic Appenin mountain range and it's a sight to see! Abruzzo is one of the wildest regions, and on the way, you will be able to see why it's one of the few untouched areas in Italy. It will be hard not to want to get off the train! Important stops along the way are Avezzano, Sulmona and Chieti. Taking the train from Pescara to Vasto is different are the train is following the coast, going through the different sea towns along the way and you'll see a lot of the Big Blue!
Now... the train station is the in an odd place. No doubt by wanting to do some savings and not have to build two stations for Vasto and neighbouring San Salvo, the train station has been built just right in the middle! Which is not practical at all for the traveller! Hopefully, you will be able to catch a bus to town or you can take a taxi (go to the station's bar and you'll find the number).
For more information about schedule and price:http://www.trenitalia.com