High speed train is a national pride of China and rightly so. It is the longest high speed railway in the world and is the most extensively used. You can easily go to anywhere in China within a day and many locals prefer taking the train to flying between cities. Some websites provide services for you to buy train tickets online but I find that buying tickets in China is actually not too difficult. Unless you are travelling during the National Day Holiday (early October), you should be able to buy your train tickets without problem.
Buying your tickets
Chinese locals can buy train tickets through WeChat (yes, again). Tourists, however, must rely on either the train station or local travel agency. If you live close to one of the high speed train station, you can easily line up at the station and buy your tickets. Otherwise, you can buy your tickets from one of the authorized train ticket dealers (火车票代售点) or travel agencies (use popular ones, like Ctrip Travel Agency 携程旅行). They typically charge a small processing fee.
You will need a passport for every passenger and their names on the ticket must match the ones on the passport. This is important because you'd need to show your passport when you present your ticket at the station. Each ticket is also assigned a car and a seat.
You can use this website to find the locations of authorized train ticket dealers.
Pro tips
If you are travelling to multiple cities and your schedule is relatively fixed, you can avoid the hassle of finding the ticket offices in every city by buying all of your train tickets at once. If your schedule changes later on, you can still change the tickets at the railway station. There is a nominal charge associated with the change though. Each ticket can only be changed once.
Train stations
For large cities, there are typically more than one high speed train stations. Choose the one that is closest to your final destination and fit your schedule.
In addition to the typical security check in China, you will also need to present your passport and your ticket when you enter the station. This way, the train company prevents solicitors from entering the station. There are enough eateries and shops inside the train stations to keep you busy while you wait.