Hangzhou East Railway Station is not the main train station. It's the dedicated high speed train station. You can get there by subway line 1 (red line) or line 4 (green line)
Suzhou Railway Station is the main train station and is very close to the old town by taxi or subway (subway line 2 or 4). Suzhou North Station is farther but since it's on subway line 2 it's quite easy to get into the old town area (40 minutes by subway)
I've never had issues getting the tickets on the day of my trip, but if you're travelling around the national holiday week (October 1st) or Chinese New Year, you should book your ticket in advance.
Train tickets can be purchased in advance at any train station in China
Tickets are be bought up to 30 days in advance
Payment is by cash
You need to bring your passport
You can check schedule or buy tickets online at trip.com, which is the English language website of Ctrip, the largest online travel booking site in China