I have a similar situation, this is what I do:

For airline check-in purpose – use both passports to verify qualification for entering the destination country

When entering Taiwan @ immigration :
1) For stay more than 30 days, use Taiwan passport (obvious ,since you have no Taiwan Visa on ur US passport)

2) if less than 30 days, you either can use Taiwan passport or US passport. Taiwan goverment gives 30 days stay permit - free of charge to US passport holders. But immigration may request you to show him the return ticket or ticket heading other ....ry as a prove you are leaving in 30days .

When leaving Taiwan @ immigration:
1) use Taiwan passport only
2) use both – you will get exit stamps for both passports.

This could be helpful when the next country you visiting may ask for exit stamp...eg:

“Dual nationality is not recognised in Malaysia and those found holding two passports of different nationalities might be refused entry; it is advisable to enter Malaysia on the passport on which you exited your last country of departure”