You can also use trip.com to book train tickets. But yes, in China, rail tickets go on sale 15 days in advance. That said, unless you're travelling during a peak holiday period, generally there aren't problems especially for big cities like yours. I used to use trip.com on my first trip but have gotten the 12306 app direct and just pick and poke my way through the chinese since I can barely read (I know, shameful being ethnically chinese). I find it quick and easy to use once you go through the setup process.

In mainland China, your ID is your ticket. You scan your ID (passport in your case) to enter the security controls at the train station, and then again to board. As a strong advice : remember aerosols >150ml are prohibited on public transports. Being a guy that shaves, that means I travel with the travel size shaving crème- learned that the first time. This applies to things like hair spray and cologne too.

I'd fly between Shanghai and Xi'an 西安. Plenty of flights between SHA and Xi'an XIY, and since I am assuming you have status with DL and thus Skyteam, that helps a lot in Asia. There are hourly flights on MU and a lot of them on the C919 so you'll have a chance to try that out . Just counter intuitive to many westerners, the cheapest domestic mainland tickets are often released 2-3 weeks before departure.