Beijing Tian'anmen Square

Tiananmen East Station and Tiananmen West Station are two nearest stations to Tiananmen Square.
The best and most convenient is to take the taxi, because there is only about 4 kilometers between your hotel and Beihai Park. If you want to take a bus, you can take the bus No 10 to reach Tiananmen West Bus Station and then change to special line 2 to reach Beihai Park. If you want to go to Wangfujing from the Park, you can take the special line 2 to reach the Qianmen and then change to bus No. 59 to reach the Wangfujing.
As for the subway, no matter what kind of subway line you take, you still need to take the bus to reach Tianqiao Bus Station, where the natural history museum locates, so I think the taxi is suggested.