Yup, Prestwick Town is the closest railway station to the centre by about half a mile. Trains run from there to Glasgow Central every 30 minutes.
From Glasgow Central the train you require is the Ayr train, they usually leave from the platforms at the far side of the station. Again, they're every 30 minutes.
It's exactly a mile to the centre from Town railway station so either a walk or the taxi is good. Bear the weather in mind though of course - so a taxi maybe your best bet.
A taxi will cost about £2.50 or £3.00.