You can only avoid US taxes if you drop citizenship altogether. US has citizenship based taxation and you are stuck paying US taxes regardless of where you reside and what other passports you have. So getting another passport (as opposed to residence permit into your US passport) does not help you at all there. Almost all countries have double taxation treaties with US so you can credit your US taxes paid against your tax obligation in your country of residence, but if their taxes are lower than US taxes, you just pay US taxes. Again, no difference if you have that other country's citizenship, or just legal residence as an American. Difference happens only if you actually renounce your US citizenship and thus cease to be a U.S. person for tax purposes.

People are getting second passports as a fancy object, and to avoid discrimination in countries where anti-Americanism is popular.