Hi, there is direct train to go from beijing to guangzhou. it may take 20 hour at least. The price is: hard seat:253 yuan, soft seat:402yuan, hard sleeper: 428yuan, soft sleeper: 675yuan. As you have two children, i think you'd better buy a hard sleeper. Or it will be quite suffering for two children to spend nearly one day in the train with a hard seat. If you go there by plane, it will cost you 3 hours, and the price is 600 yuan.
My suggestion is you'd better take a plane. Because the price of plane is not very expensive. What's more, this is the first time for you to go to China, you'd better make the first journey comfortable!
Oh, i nearly forget to tell you that it is impossible to purchase the train tickets before you leave US.
Welcome to China!