Connecting in PEK is just terrible. The connection experience is awful, and many domestic flights within China are delayed. I fly PEK SHA 3-4x/month and >50% are delayed regardless of which airline. It has to do with the military. If you fly to PEK and stay for a day or two, you could take the train to SHA. It's about 5 hours, so when you count check in times etc, about the same as flying. When you count delays, it's usually better. All my local friends here take the train. I still fly but I do it for the miles and have gotten good at rolling with the delays and keeping myself productive.
I live in PEK and today the sky is blue. Hope you come on a blue sky day, otherwise, bring a mask.
You can't beat the great wall - it's worth seeing at least once in your life. Probably best to just go a tourist area if it's your first time. Go to Mutanyu and take the toboggans down the mountain. It's pretty fun.