Originally Posted by RadioGirl

There have been numerous threads in this forum before on this subject, which would give you a lot of advice.

At each transaction, think about why the person needs to see your passport, and choose accordingly.

Entering a country, the immigration people need to know you're allowed to be there. Show the passport of the country you're entering.

Leaving a country, the airline check-in person is making sure you'll be allowed entry to your next destination. (They really don't, or shouldn't, care whether you're in their country legally at that point.) Leaving the US to go to Taiwan, show your Taiwanese passport to the airline person.

Leaving a country, exit immigration/border control (in countries that do that) needs to know you've been there legally. Show the passport of the country you're leaving.

At security checkpoints in the US, they just "need" an ID that matches the boarding pass. Use either passport.

That said, having two or more passports is not uncommon nor (in general) illegal, so there's no problem with anyone finding out that you have a second passport. It all feels very James Bond the first time, but you get used to it.