One week not enough.
Great wall two spots, the usual low difficulty one at Batalin, another one at Mutianyu (takes one thousand steps up hill then you reach the great wall) very scenetic, can even listen to the wind.
Summer palace, one day.
Forbidden city (make sure you enter from tian an men enterance and walk back, if you enter from rear, the sight is different and sun gets in your eyes in the morning) and tiananmen square; total one day
Xiang San (fragrant hill, can see red leaves in Oct) , take cable car up and see whole of Beijing skyline. 3/4 day.
Ming tombs, one day because there are afew tombs, plus the entrance/ passage way to all the tombs. There is one main one, the rest are not really fully opened. The main one, you can go underground, 5 levels down and see the way the coffins are laid, and enjoy the garden at ground level. Nice and quite. Then go back to the passage way with all the stone figures (passage to the good 'fengshui' burial grounds'.
Hou Hai, like clark/boat quay, visit at night, pubs and eatries. Must visit the small lanes very nice gift shops at night.
BuxingJie. Shopping district near tianan men, can go after visit to tian an men. Go BuxingJie to eat dinner and rest.
Yue tan, Tian tan, Ri Tan in city, one day among them.
There are also many surrounding sites around the city, like the sliver pagoda, the thousand bamboo forest, caves etc. Different difficulty level.
Four star hotel cost you about rmb300 to 400, corporate rate. No corporate rate, rmb700 to 800 (cut throat)
Transportation to great wall plus toll included, about rmb 400 to 500, depending on type of vehicle and number of ppl.
GuwanChen, (antiques sales city), 1/4 day to 1/2 day in city.
If at tiananMen, if you go visit museum, another one day.
If you also visit military museum, another half day.
Many more, but I sleepy liao. If you got question, please ask and I help answer if I can.
Thus, far all duration quoted are quite rush one. Ming tomes are nice tranquil places to be enjoyed.
March and early april good time. But late April and May is Spring time. So if you go in April many not see any nice flowers. But if you go during spring, there are alot of pollens flying around and your skin will feel very dusty like minute cotton sticking to your body hair.
Meals not a problem, alot of Macs and KFC, etc.
Sichuan, the nine gougres and yellow dragon river is very nice. But not very credit card friendly. But very nice, very very nice. The multi colour lake, I feel like I want to go again,...sob sob....memories.....aarrhh....