For the kind of routes you mention I would recommend booking at least the day before, and maybe a couple of days beforehand, especially if you want to take the CRH high speed trains (Suzhou is now under 30 minutes from Shanghai, Hangzhou is also faster now).
That said, if you are unfussy about the time of day you travel (i.e. not at rush hour) then you can often get tickets on the day of travel, though it is unlikely you'll get tickets for a train leaving immediately so may have to come back to the station a couple of hours later. Equally, however, I have been in situations where I needed to travel NEXT day for work, and was unable to do so because all trains before 4pm had already sold out between Shanghai and Suzhou.
Also bear in mind that some trains now only go to the new high speed stations at Hongqiao Airport and Shanghai South, rather than the main Shanghai station.
To buy tickets, I would get your hotel to do it (if staying at a hotel). They will charge a couple of dollars for admin but it will save you time and hassle. The tickets themselves are very cheap, so soft seat (First Class) is an affordable option.
If you want to DIY, then some travel agents will sell you train tickets, and there are boking offices located around town.