Hi, at present there is no subway running to Yiheyuan or Yuanmingyuan. Good news is that Beijing Subway Line 4 including these two stops will put into use since Spe.28. THAT IS TO SAY, you can take Line 2 from Chongwenmen to Xuanwumen and transfer to Line 4 from Xuanwumen to Yuanmingyuan or Beigongmen (North Gate of the Summer Palace).

Mr.Norm. From Yuanmingyuan, how can I get to the Olympic stadium & Water Cube? By subway or taxi? Thank you.

Three options for you to choose:
1> Taxi, CNY30
2> Subway, CNY2, 2 transfers. first take subway line 4 from Yuanmingyuan to Haidianhuangzhuang, then transfer to line 10 from Haidianhuangzhuang to Beitucheng, at last take line 8 from Beitucheng to Olympic Sports Center
3> Bus, CNY 1.5, Bus No.696 from the south gate of Yuanmingyuan to Asia Game Village (passing 9 stops)

Mr.Norm. Is the Asia Game Village same place as the Olympic Sports Center?
Which mode of transport is faster - subway or taxi? Thank you.

There are near each other. But subway don't extend to Asia Games Village. Just take taxi. It will cost around CNY10.

After the Bird Nest, which is the best route to return to Wangfujing? Thank you.

Suggest you to take the subway which is muchd faster than city-bus. And, you could find the subway map to show your direct easily in every subway station. Or, please refer to the subway map at https://www.travelchinaguide.com/images/map/beijing/metro-planning.jpg