It is suggested that you take a flight to Xian since the train will take you about 36 hours although it is cheaper. The real-time flight info could be searched at https://www.travelchinaguide.com/china-flights/

From Xian to Beijing: you could take a night train, taking about 11~12 hours on the way. The ticket fare is about CNY270/hard-sleeper and CNY400/soft-sleeper.

If you're going to have a budget travel, I think CNY8,000 is enough. However,if you would like to participate in a group, it will be higher but the tour will be much easier for you.