Travel from US

Hi,
I am a UK and US citizen and planning to travel from the US to Chengdu with connecting flight in Shanghai (stay in Chengdu less than 48 hours) then on to Nepal and Tibet (with acquired travel permits via group tour in Nepal and Tibet). I then plan to fly back to Chengdu from Tibet (staying in Chengdu less than 24 hours) and then return to the US (with short layover in Shanghai - not leaving the airport). What type of entry permit is required and which passport would you recommend I apply for the visa with? Thank you!