If you wish to be in Spain for longer than 90 days every 180, you must apply for a visa before you enter the country. There's a variety of visa options for both short and long-term stays. Read about long-term Spanish visas.
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With a valid U.S. passport, you can stay up to 90 days for tourism or business during any 180-day period. Do not overstay!
Luckily, when you've applied for your TIE card and are waiting for it to arrive, you can leave Spain by applying for Autorización de Regreso, an authorisation to return.
Furthermore, once you've used up your quota of 90 days, you cannot return to Schengen until 90 more days have passed. For example, if you enter Spain on January 1st and spend 90 days in the country until June 30th, you cannot return to Spain until at least the end of September.
Generally, Spain does not permit dual citizenship with another country other than with Iberian countries. If you obtain Spanish citizenship, you will have to renounce your original nationality. There are exceptions, including Spanish-American countries, Andorra, the Philippines, Equatorial Guinea or Portugal.