China has three distinct concepts of homeland: 籍贯 (jiguan)，故乡 (guxiang) and 老家 (laojia). Many people in traditional families will be headed to their 籍贯 - their paternal grandfather’s hometown. 籍贯 is the only one of these three used on official documents like school records or hukous. 故乡 (lit. former hometown) sounds a bit more poetic, and can be used to refer to any place with which you have developed an emotional attachment (Nanjing would now be considered this columnist’s second 故乡）. 老家 (old home) is very colloquial and usually refers to the place where you grew up or where your parents live. Most people will probably head for their 老家 for the holiday.
As it happens, the reason tickets are hard to buy at this time of year is not entirely because everyone in China will be traveling during the same few days. Scalping of train tickets by people called 黄牛 (yellow bulls) has historically been a huge problem, with people using their connections to buy obscene numbers of tickets and selling them to desperate migrants returning home to their families with the money they’ve earned working. The government has taken huge steps to curb this in recent years, but the problem is probably impossible to totally eliminate.
Note for foreigners coming to China: be aware that going home for the holiday with your S.O. strongly implies that a wedding is not far off. Even if you’re just friends, it’s probably a good idea to bring someone else along if you’re visiting the hometown of someone of the opposite sex during Spring Festival to avoid adding grist to the vicious hometown rumor mill.