Are you headed for the town of Teramo or somewhere in the province? The town itself is reachable by train so that's probably a good option. The eastern coast is separated by a mountain range so the tracks tend to run north/south up and down the coast. Bologna is the hopping off point for the coast so whether you leave from Florence of Venice you will probably end up getting there through Bologna if you take the train.
Venice > Florence > Teramo is a pretty direct way to get there since Florence is a bit closer than Venice.
Driving from Rome is an option but unless you're really comfortable driving in Italy I would suggest getting somewhere outside of Rome and closer to Teramo before renting a car. Maybe train to Ancona and then rent a car if you feel you have to?
You will traveling a good length of the east coast of Italy so maybe look for beaches there? Starting at Rimini and down the coastline is a whole series of small towns. You should also be clear on whether you want a "beach town" or a coastal town. Please like Sorrento and Capri are ocean side coastal towns but have little in the way of actual sandy beaches.
I wouldn't fly internally anywhere in northern Italy. The train from Rome to Venice is 4.5 hours. I don't think you can beat that with a flight between getting to airport, showing up early and then getting from the airport back into the arrival city. Just take the train and leave out all the airport drama.
Have a great trip,
=Tod