Alll schedule info is available on www.sbb.ch/en
Allow a day if possible and you can visit both sides and Schaffhausen as well. Or if not, then a morning. I personally prefer the Neuhausen side, but others like Schloss Laufen...so just pick which makes sense from a time-table poitn of view. Journey time on the train from Zurich is about an hour...
I just want to add, that you can easily walk from one side to the other over a walking path on the railway bridge.
From Schloss Laufen you are very close to the fall and it is very impressive. By the way, there are great restaurants at Schloss Laufen and Schloss Wörth (on the other side).
The old town of Schaffhausen is not very large but I am sure you will enjoy it.
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