The airport shuttle bus from Friedrichshafen stops at the Messepark on the outskirts of town. The other option is by train; many intercity trains running through Austria stop in Dornbirn.
Local and suburban trains operated by OBB [http://www.oebb.at] 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.
Go for a hike, best visit the Scouts Place on "Zanzenberg". Even more interesting is to take the "Stadtbus" or "Landbus" to Rappenloch, walk straight into the Rappenloch (30mins), 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" Two cheap Restaurants or at least Reststations are on that track.
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 "Grune 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 euro 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.
Fachhochschule Vorarlberg [http://www.fhv.at]
Every 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 veg, organic beauty products, flowers... It's nice to spend a while wandering round it, and if you buy something you can be sure it's locally produced and therefore better for the regional economy. Everyone's a winner!
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. Mr John's is an upstairs bar in the corner opposite the church where you can smoke a water pipe if you so wish. For coffee go to Klappe, a movie-themed cafe bar a little further up the street.
7er bar in Eisengasse is the true alternative scene, with live rock music and other events. They even sell beer! It's often packed and is a great place to have a quick drink or spend the entire evening.