Should I book my train connections in advance in China?

I will be travelling in China this July. My trips will be

• Beijing - Shanghai
• Shanghai - Hong Kong
• Hong Kong - Tianjin

I have read a lot about the train situation on Seat61 and am thinking about doing the following. Please consider the timeframe of July.

• Beijing - Shanghai via hard sleeper, buy the tickets when we arrive in Beijing at a train station. If the hard sleeper is sold out, we'll fall back on the high speed trains, which I assume should have two seats available at short notice, as they leave quite regularly. Is it realistic to buy these tickets a few days beforehand if we're flexible on exact times or does this connection sell out completely on a regular basis?

• Shanghai - Hong Kong via hard sleeper. I am considering buying these tickets beforehand online as I imagine they may get sold out easily. Is it realistic to buy these tickets on-location or should I expect them to sell out?

• Hong Kong - Tianjin via hard sleeper from HK - Beijing. I am considering buying these tickets beforehand online as well for the same reason. I imagine Beijing - Tianjin is easy and can be arranged when we arrive in Beijing for the second time. Same question as the Shanghai - HK connection

Update

For future travellers, we bought all our tickets after we got to China. Booking them online or via agency/hotel is a good idea if you want to avoid the hassle of getting them at the station. We were lucky at the station as our sales clerk spoke a few words of English. If you want to get them at the station, print out the form supplied by Seat61 under the chapter "How to buy tickets at the station".

The sleeper trains may sell out and we only got a seat (instead of a bed) for our Beijing-Shanghai connection, while the HK-Beijing connection still had a lot of empty beds. Book as far in advance as possible though if you want to make sure you get a bed. The bullet Beijing-Tianjin connection was easy, we got it a half hour before boarding the train, though the train was pretty full. There are also slower trains that go all night.

In general:

• get to the train station a half hour early for any connection
• if you need tickets, add at least another half hour, as there are always tons of people queueing up at the ticket counters and you'll have to show your passport when buying tickets
• get there an hour early for any connection leaving mainland China as you will have to go through Customs and Passport Control - they close this 20-30 minutes before the train leaves, so you may miss the train even if you get there 20 minutes early!