By plane
Fuzhou has air links to most other major Chinese cities. The airport is an hour from town in the suburb of Changle, ¥20 by bus. You can get the bus at the Apollo Hotel. There are also complimentary shuttles to the Apollo Hotel from other places such as the Min Jiang Hotel. Shared taxis also go from there, at around ¥25. A private taxi would be at least ¥100, likely more unless you haggle well.
The airport also serves international flights to Hong Kong, Singapore, Osaka and Taipei.
By train
The main Fuzhou train station is in the northeast of the city. You can reach it via #5 or #22 city busses, or it is cheap and quick by taxi from anywhere in central Fuzhou. There is also a new Fuzhou South station, located rather inconveniently. A taxi to it will be ¥50 or so from downtown and take about half an hour. The K2 bus runs between the two stations; the cheapest way to the South station if you have time is to go to the main station and take that bus from the parking lot off to your left as you face the station entrance.
A new fast train between Xiamen and Fuzhou is available, leaving from the main Fuzhou station almost hourly, for under ¥100 one way.
There is also a fast train from the South station to Shanghai via various towns in Northern Fujian, Wenzhou, Ningbo, and Hangzhou. Around 6 hours, ¥282 for second class, 330 for first.
By bus
Direct overnight buses to/from Hong Kong, Shenzhen or Guangzhou exist, with sleeping bunks. ¥220-350, it is usually worth the extra for comfort. Buses from Xiamen are ¥70-90.