To legally teach in China, you will need a Z visa to enter the country and to begin the process of obtaining a resident permit.
To legally teach in Korea, you will need to get an E1 or an E2 Visa. These are legal working visas for English teachers. English professors working at educational institutes higher than a junior college, need an E-1 Visa while those working as conversation instructors, require an E-2 Visa.
To legally teach in Japan you need to get a Working Visa and a Certificate of Alien Registration.
To legally teach in Taiwan, you will need to obtain a Work Permit and Alien Resident Certificate (ARC).