Do I have to pay additional fee if I'm bring an extra luggage?

Hello there! My parents and I will be traveling from overseas. My question is, will it be a problem if we have additional baggage? I understand that there's a 20 kilo per adult luggage allowance. Will it be a problem if we have an additional luggage with us? Do we need to pay extra for it? We'll be riding a high speed train from Shanghai Hongqiao to Hangzhou, then Hangzhou to Suzhou, and lastly Suzhou to Shanghai.
Is the hand-carried luggage counted as part of that 20 kilo luggage allowance? Also, you mentioned that you need to check-in the luggages 2-3 days before departure. We won't be able to do that. Will checking-in luggage take that much time? What time do we need to be at the station if the departure time is at 1:30pm?