Friedrichshafen Airport (IATA: FDH)  is the smaller (and closer one). It connects with direct flights from Dublin, London, Berlin, Prague. There is an airport shuttle bus from Friedrichshafen Airport, which stops at the Messepark on the outskirts.
Memmingen Airport (IATA: FMM) is the main connection with Europe besides the bigger airports of Munich and Zurich, which are notably farther away.
There are many intercity trains going from Lindau (Germany) to Innsbruck and farther away that stop in Dornbirn main station (Hauptbahnhof).
Dornbirn has plenty of options regarding public transportion, considering its relatively small size. Nine different Stadtbus  lines operate inside the city and several Landbus  lines connect Dornbirn with its neighboring towns.
Local and suburban trains operated by ÖBB  stop at the main station and the three suburban stations (Hatlerdorf, Schoren, Haselstauden). Intercity and Eurocity trains stop at the main station which is also a central hub for bus lines.
From the main station, it takes about 10 min walking to reach the city centre, which is fairly small. However, many attractions and museums are outside and you might want to take a bus.
Dornbirn is mainly interesting when considered from a nature perspective. There are some museums like the Inatura or the Rolls Royce Museum, but where the city shines is in its beautiful surroundings.
From the main square, there is a 2 hours hike to the peak of the Karen, the mountain that oversees the city. It is a fairly advanced hike, not recommended for begginers, but definitely worth the time for the sporty. If in doubt, there is a cable car (Seilbahn) going to the top for approximately 10€.
Another option for hiking is to take the "Stadtbus" or "Landbus" to Gütle, the last stop on the route. Once there, walk straight into the Rappenloch (30 min), pass the Lake, continue to "Alploch" and then up to the "Kirchle", a great natural halfcave, impressive to look at. You can even continue and watch the strange geologic formations on side of the "Ebniter Strasse". There are at least three restaurants/kiosks on the way, with bathroom and benches to rest.
Go for a hike, best visit the Scouts Place on "Zanzenberg".
On those warm summer nights it's great to go to the central square of the town. Especially Thrusdays and Fridays can be recommended. During Advent a large Advent market is held in the market place and surrounding streets and squares. In the evenings the Gluehwein stands stay open and the atmosphere gets quite merry and fun. Another great time of year is Fasching, the run up to Lent, when there are often parades and bands in the market place.
Fans of alternative Music watch out for several locations: Visit the "Grüne Pavillion" close to the Central Square and on Weekend you can also use the free Shuttlebus taking you to then #1 Nightclub, the "Conrad Sohm", twice an hour, round midnight. Also got to the "Innauer" for great drum'n'base and dance. To hang out you can visit the "Tivoli" Whinebar or eat great italian Pizza at "Paolo and Andreas", charging only €5 when taking away and located close to the center as well.
For cultural events (art-house films, music, dance) the Spielboden in the district of Rohrbach is worth a visit. There are a number of museums to visit; Inatura is the interactive museum about the natural world which will appeal especially but not exclusively to children. Near Inatura in the Stadtgarten is Kunstraum Dornbirn, where you can see contemporary art around the theme of "art and nature". The world's largest Rolls-royce museum is out of town to the west - take the number 5 bus as it's a bit of a trek.
The Stadtbad (indoor swimming pool) is worth a visit for the modern architecture alone. The facilities are brand new - there are 3 pools, 2 diving boards and a water flume as well as a sauna. Closed on Mondays.
The Fachhochschule Vorarlberg (FHV)  has a remarkable library and services like printing and internet access.
Dornbirn has a lively city centre with a large pedestrian zone and many shops. If you can't find what you are looking for inside the city, there is also a large mall on the outskirts.
Every Wenesday and Saturday, a farmers' market is held in the market place. There are stalls selling anything you could possibly desire; cheese and meat from small farms in the Bregenzerwald, socks, fresh bread, fruit and vegetables, organic beauty products, flowers... All the product are of the best quality and locally produced.
Try Sushi at 21, a classy restaurant on the Marktplatz. They also do a great selection of teas.
Alternatively, go to the Karren panorama restaurant, which perches on the side of a mountain above town. They serve mainly good typically Austrian cuisine! You can see it from everywhere in the city. The view over the Rhine Valley and Lake Constance is stunning night and day and it's often sunny up there when it's foggy down below. They do a very good value cable car and meal combination ticket on weekday lunchtimes. To get there take the number 5 bus from the station to the Karrenseilbahn (cable car).
The market place is the centre of all activity with a number of cosy cafes and more lively venues.