How to go from Beijing to shanghai the easiest ways and fares and how far and time duration ?

The distance between these two cities is about 750 miles.
Taking the direct flight is the easiest and fastest way for you to choose. It just take around 2.5 hours on the way. In usual time, the ticket fare is CNY 600 to 700 for the single trip.
Or you can travel by bullet trains. Duration is about 5.5-6 hours. Ticket fare for a second class seat is CNY553.

If you travel on a budget, you may consider T109/T110 between Beijing Railway Station and SH Railway Station.
Travel Time: 15 hours
Ticket Prices: CNY315.5 for hard sleeper and CNY497.5 for soft sleeper.

One more thing - find a good China travel guide app for your phone. It can help you find your way around and also point out great things to see and do.