Researching family/genealogy records in Taiwan and China

I think your solution will need to consist of several plans, probably executed concurrently:

• Interview as many living relatives as you can -- they can't all have exactly the same knowledge or attitudes. The information on individuals may be inconsistent, but, just like many apocryphal family stories, there will be nuggets of truth. If there are written sources like letters or diaries, include them in your research.
• Find primary and non-subjective records -- official records of all types. Start with documents found at home. Use the genealogy websites (FamilySearch.org and ancestry.com, which has a Chinese affiliate), but, depending on access and expense, you will eventually have to look at non-digitized records at archives. Learn what records were generated and what was preserved.
• Educate yourself on the history and geography of your ancestral regions. Start general, but get as specific as you can. Include the recent history that influenced your living relatives' attitudes. Books and websites and even blogs can be helpful, or take courses.

Record who told you what, and where you found which data. You'll want to be able to judge reliability and probably to re-examine details.

Your question is very broad but when starting to investigate genealogy and family history in a new country a great place to look is the FamilySearch Research Wiki:

• Taiwan: https://www.familysearch.org/wiki/en/Taiwan_Genealogy
• China: https://www.familysearch.org/wiki/en/China_Genealogy