Quanzhou's airport is at Jinjiang and with a rather limited flight schedule, especially when compared to Xiamen. Therefore, it's most possible that travellers will be flying to Xiamen airport and then changing to land transport to get to Quanzhou.
Xiamen airport is north of the city. No need to go to town to get a long-distance bus to Quanzhou (or to a lot of destinations in the province) as the long-distance buses leaving from town will have to pass by Xiamen airport anyway. The airport has a counter that will arrange for transfers to these long-distance buses. Buy your ticket here; make sure to say that you don't want insurance if you don't, otherwise they'll automatically include it in the price. Ticket in hand, walk out of the arrivals concourse, turn left and continue walking past the KFC. Look out for the signs along the way.
There will be a minibus to ferry passengers from the airport to a shady-looking (but which isn't) transfer point a little way from the airport. The staff at this transfer point will yell (announce) the destination and departure time in Chinese when each bus comes in. Listen for your bus (and the correct departure time) carefully; if you don't understand Chinese I suppose you could shove your ticket under the nose of a staff member and impress upon them to drag you over to the correct bus when it comes.
The ride was reasonably comfortable though nothing to sing about. I think I took about 3 hours in all from Xiamen airport to Quanzhou, with 2 hours of that spent on the road.
Bus to Chongwu cost 10 RMB; there's no timing specified on the ticket, but rather a latest time that you must use the ticket by -- mine was 1830 so I assume that's the last departure time. If you have to go via a change of buses at Hui'an, it'll cost you a few more RMB.
Luxury express buses to Shenzhen (Shenzhen Qiaoshe) leave between 0900 and 2130.
Express buses to Fuzhou (Minjiang Hotel or Fuzhou north long-distance bus station) leave hourly from 0700 to 1600, then 1730 and 1900. Price is 55 RMB.
If coming from Fuzhou, the best buses leave from Fuzhou's south long-distance bus station. These buses are luxurious, with leather upholstery and the works. The earliest departure is 0700 and they cost 63 RMB. Journey time about 3 hours.
Quanzhou's train links are woefully underdeveloped, and there's no really convenient airport nearby. So if you're planning to visit, I'll bet that you'll have to come and leave by long-distance bus.
The centre of all long-distance bus transport has now moved to a spanking new long-distance bus station to the SE of downtown Quanzhou, you'll have to take a taxi or bus (sorry, forgot which number it was) to town. Destinations served are pretty comprehensive.
Alternatively, there're also other private (?) long-distance bus operations in town that you could use if you don't want to trek all the way out there.
The Taiwan Museum (for want of a shorter name) and Quanzhou Museum are located in a rather inconvenient re public transport. The road that it lies on was totally devoid of any public transport at time of visit, not sure if the situation has improved now. You could either take a taxi or walk a very long way to any bus stop.