Hi, Badaling and Mutianyu both are worth seeing during October. But since you will visit the Ming tombs after the Great Wall, I suggest you to visit Badaling for they are located very near and the transportation is easy. Mutianyu and Ming tombs are in opposite positions that you have to spend two days to visit both.

Thanks for your suggestion. I'll be staying near Tiananmen Sq. How can I get to Badaling from my hotel & what time do I have to set-off in the morning so I can have time for Ming tombs in the afternoon?? Thanks again!!

You can take bus no.5 at the east of Tiananmen Square and get off at the terminal station - Deshengmen. At Deshengmen Station, take bus no.919 to badaling Great Wall. The 919 bus ticket costs CNY 12 by cash and CNY 4.8 by Beijing public transportation card. Why I have to tell you the price, because you should remember this price and check it before getting on the bus, because there are many private buses which claim to be 919. Or you can visit this page to avoid taking a wrong 919 bus: https://www.travelchinaguide.com/cityguides/beijing/greatwall.htm

Yes, you should start early to visit this two sights, maybe 7 or 8 in the morning, and spend two hours on the wall, and visit Ming tombs in the afternoon. Many people prefer to arrange their Badaling-Ming Tombs travel in this way. Good trip!

Thanks for the useful info.
From Badaling, how to I get to Ming tombs? How long should I spend at Ming tombs? After Ming tombs, how can I get back to Beijing? Will I have enough time to visit Marco Polo bridge when I get back to Beijing? Thanks again.

does the Beijing public transport card comes in 1day or 3days unlimited travel for tourist? thanks.

You can take the bus no.919 (慢)(means slow line) from Badaling Station and get off at MingHuang LaXiang Guan Station (The Dynasty Wax Works Palace), and take bus no.314 and get off at Ding Mausoleum (one of the thirteen Ming Tombs).

Two hours are enough to visit Ming tombs. You can take bus no.314 from Ding Mausoleum (one of the 13 Ming tombs) and get off at Minghuang LaXiang Guan Station (The Dynasty Wax Works Palace), and take bus no.919 (慢)and get off at the final station - Deshengmen Gate, and when you arrive at Deshengmen Gate, take bus no.5 and get off at west Tiananmen Square.

Sounds complicated? but it is the most easiest way travel by bus.

Beijing Public Transportation Card is only a card which you store money in it and it can help you travel around Beijing City easily and cheap. There is no limited days. I think this page will help you understand what a smart card it is.
https://www.travelchinaguide.com/cityguides/beijing/transportation/smart-card.htm

Probably you don't have enough time to visit Marco Polo bridge. Because Marco Polo Bridge is located too far away from Tiananmen Square and it is already closed at 19:00 in October. And you will be get tired after visiting Badaling and Ming Tombs. So why not just get a good rest and enjoy the night view. On the next day, visit Marco Polo Bridge. So you will not be pressed for time.

Thanks all so much for the useful info.