Can I get a train direct from Pudong airport to Suzhou or must I get to the city centre first?

There is no direct train from the airport to Suzhou, you need to take the airport shuttle bus line 5 to SH railway station, where you can take the bullet train.

By the way, just in the airport, there are direct long-distance buses you can take. After 4 pm, the schedule is as follows:
16:50, 17:20, 18:10, 18:50, 19:40 ,20:30 and 21:00.