Train is the way to go. It is roomy and high-speed. Take the train!

Bus: looking at around a 2 day+ journey

Plane: This is China and delays are more than likely. Taxi from Chaoyang to capital airport 80-100rmb, taxi from Pudong to city centre 100rmb+. 

Train: City to City  and as Fruit said it is high speed rail and little to no delays and both stations are connected to the subway line. 

They used to have a nice train service. It was called the 'Z Train'. Very nice, like a rolling hotel room. You go in the evening and wake up in the morning in the city. A nice way to travel between BJ & SH.

Train is great if you need time out and have plenty of time  however for shear convenience the plane is the best option

If in a hurry and money is no option, then use airplanes.  If you have some time, between bus and train I recommend train, room to walk, have rest rooms available, and less prone to accidents.  Bullet trains are maybe 200 Rmb lower than airfares I heard.  Using buses you must sit all the time, and stops for 20 minutes every two hours or so.

for the money but it might take a little longer walk or bike

I took the bullet train three weeks ago from Beijing to Shanghai. I travelled second class for 555 RMB and it was much cheaper, more comfortable and convenient than by plane. The seats are bigger and has much more room than economy class on a plane. It is also clean with washrooms and safe.

I took the bullet train three weeks ago from Beijing to Shanghai. I travelled second class for 555 RMB and it was much cheaper, more comfortable and convenient than by plane. The seats are bigger and has much more room than economy class on a plane. It is also clean with washrooms and safe.