72 hour exemption or multi entry tourist visa?!

We're all UK passport holders. 2 adults, one 2 year old child.

We're traveling to Taipei via Shanghai. My flights are London -> Shanghai, stay in Shanghai for just over 48 hours, flying Shanghai to Taipei, staying in Taiwan for 12 days, flying from Taipei to Shanghai, staying 24 hours and then flying back to London.

I have no idea which visa I need or whether I can get the 72 hour exemption. I can't tell if I can get the 72 hour exemption or whether I would need a multi entrance visa for entering mainland China Twice.

The websites are so confusing and I am having a hard time figuring out what to do. I really don't want to have any dramas on a family holiday so I would really love some help if anyone has any ideas!

If I can use an exemption - do I need to register somehow?

Thank you