By far the closest airport to Penne is Pescara. Pescara has direct flights from Stansted in the UK and from Toronto in Canada, as well as flights from some other European airports and internal flights from Milan. You can see the list on the Pescara airport timetable. If these airports aren't convenient for you - and you don't want to change in Milan - an alternative is to fly to Rome's Fiumicino airport and travel from there.
The quickest way to get from Rome to Pescara is to pick up a hire car at the airport and drive. This takes 3 - 4 hours, epending on how fast you drive. You need to go half way round the Rome ring road, exiting on the A24 motorway. The hectic bit is the ring road; after that, it's very straightforward: follow the A24 towards L'Aquila, then bear off south-east on the A25. Leave the A25 at the Chieti/Pescara West exit onto the RA12, and then the SS81, which is a minor (but good) road leading to Penne. The route from here is the same as from Pescara airport.
You can also get to Penne from Rome by train and bus, or by coach. The train leaves from Tiburtina station in the south of Rome; there is a direct train from Fiumicino to Tiburtina every 15 minutes, with 45 minutes travel time. The train from Tiburtina goes to Pescara, not Penne, and you will need to catch a bus from Pescara to Penne. Tickets for both trains can be pre-booked on the trenitalia site. This is a slow train with a picturesque route winding through the mountains. It takes 4 hours to reach Pescara. The Pescara bus station is in front of the train station, and there is a direct (but slow) bus to Penne, adding another hour to the journey. There are some more details on the train page on this site.
The coach goes direct from Fiumicino to Penne and using the motorway, so it is faster than the train/bus option (4 hours total). It runs only twice a day, once mid-morning and once late evening. The timetable is on the Prontobus web site.
Direct flights to Pescara from the UK are all run by Ryanair from Stansted. Depending on the time of year the number of flights varies from two a week between October and January to one a day in the summer. Book the flights from the Ryanair website; prices vary greatly depending on the date, and Ryanair are very strict about the amount of weight you are allowed to carry with you - be prepared for unexpected surcharges.
There is a taxi stand outside the airport; a direct ride to Penne costs around 40 - 50 euros. There is also a bus stop; all buses from here go to Pescara train station. There is another bus which runs from the train station to Penne once an hour; the bus journey from the airport to Penne takes about 90 minutes in total. But the easiest and most flexible way to travel is to hire a car from the airport (which has stands for Avis, Sixt, Hertz, Maggiore and Europcar) and drive to Penne.