In theory, these should be my go to sites. However, it all depends on topic, quality of analysis, and so on. In any case, here are five typically considered moderate (but not by everyone).
Aspen Institute (DC, with an Aspen, CO campus), until recently headed by best-selling author Walter Isaacson. It fosters "enlightened leadership, timeless ideas and values, and open-minded dialogue." Interests include foreign policy and national security issues, development of business leaders, no child left behind, and so on
Brookings Institution (DC) founded by Robert Brookings in 1916 to analyze public policy issues. Currently it focuses on social science research, especially economics. Mission: strengthen American democracy, foster economic and social welfare, security and opportunity, and a more secure international system. Brookings studied the causes of the Great Depression and later opposed many FDR New Deal policies, then assisted the Marshall Plan, contributed to creation of the UN and Congressional Budget Office.
Carnegie Endowment (DC) founded by Andrew Carnegie in 1910, with Senator Elihu Root as first president. They declared in 1917 the "most effective means of promoting peace is war against Germany." John Foster Dulles was chairman until picked by Eisenhower to be Secretary of State. Their webpage begins with "Vladimir Putin's Russia Goes Global."
Rand Corporation (Santa Monica) global policy group initially founded by Douglas Aircraft for military research. Projects included the space race, nuclear mutually assured destruction (MAD), and artificial intelligence. National security remains the major focus, such as their Homeland Security Operational Analysis Center. Note that Daniel Elsberg of Pentagon papers fame (or infamy) worked for Rand.
Urban Institute (DC) economic and social policy research, established by LBJ administration to study urban problems [suggesting this should be categorized as a liberal think tank]. The federal government continues to be the #1 funding source. Web page focuses on education funding, health insurance markets, and prisons.