You absolutely do not need to renounce or relinquish your US citizenship or get dual citizenship when you move and live abroad.
I've been living abroad for about a little over five years. Now, I live in Guatemala and have been here for over four years. I am only a US citizen. I'm not even a permanent resident of Guatemala. I've got three children and the last of one was born in Guatemala. Getting residency for my youngest child was very easy based on the local rules. However, it's not a step that I really needed to take because we frequently travel, whether we're going back to the US or going to other countries on a vacation.
Guatemala happens to have a very generous tourist visa situation where they give 90 days for Americans. If you want, you can get an extension on 90 days up to 180 days in Guatemala without having to leave, which a bunch of countries have. Panama has an automatic 180-day visa for US citizens. Nicaragua and Costa Rica allow 90 days for US citizens. It's easy to leave the country, stay out of the country for a few days, and come back in.
Residency status is certainly not needed. Relinquishing your US citizenship is absolutely not needed. I have some clients who are considering residency and relinquishing their US citizenship for tax reasons, but it could be quite problematic to relinquish your citizenship. There is a lot of paperwork involved. If you have a lot of assets, there is an exit tax associated with giving up your US citizenship that can be quite painful.
Giving up your US citizenship is really not needed for most people and 99.9% of US citizens don't need to do it to live abroad. Depending on the tourist visa situation, you may even need to establish permanent residency status in the local country you're moving to.
(People of Mayan descent in a horse race in Todos Santos Cuchumatán, Guatemala, pictured.)