There is no need for you to renounce or relinquish you US citizenship if you move abroad. The same is true for Canadians and citizens from anywhere else in the world. You can live in Costa Rica either as a perpetual tourist or a permanent resident.
Being a perpetual tourist means that you need to leave the country every ninety days, which should not be an inconvenience for someone who is a traveler anyways.
For someone who lives where I do, however, in Tamarindo, Costa Rica, which is about 2.5 hours from the Nicaraguan border, being a perpetual tourist means a simple drive up to the border once every three months, where he can simply walk across the border, get your papers stamped, walk back across the border, get your paper stamped, and be home again on the same day.
The process a perpetual tourist has to endure every 90 days takes about 6 hours – 2 ½ hours up, 2 ½ hours back and hour across the border. You can stay in Costa Rica as a perpetual tourist because by doing so, you are fundamentally invisible to the Costa Rican government, except for the fact that you get your passport stamped. A lot of people do that.
Permanent residency in Costa Rica is extremely simple. You can apply for permanent residency in Costa Rica on the basis of being a recipient of a pension of around US $1,500 a month. You can also apply for permanent residency on the basis of being an investor with an investment amount of $200,000.
If you marry a Costa Rican, then you can get a residency on the basis of family. The bureaucracy in Costa Rica is slow moving and inefficient, but one of the big advantages you’ll get from that is you don’t have to actually get the residency to enjoy the benefits of it; you simply have to apply for it. Once you’ve applied for your residency, you get a piece of paper saying that you’ve applied for it. While your residency application is pending, which may take about a year, the government of Costa Rica treats you as if you are already a resident, so you don’t have to leave the country.
There is also an option to apply for a citizenship after a few years, or sooner if you marry a Costa Rican. You do not have to renounce your US citizenship; I’ve never heard of that as being an issue.
(Map showing the location of The Oaks Tamarindo Condominiums, Costa Rica, pictured.)