Here's a step-by-step of how it happens:
Constitutional amendments
There are three kinds of constitutional amendments that can be passed. These are:
1. Congressional term limits
2. Flushing the Supreme Court
3. Reaganifying the country
(1) happens if public opinion gets high enough that 2/3 of the state legislatures agree to a constitutional convention. Generally this means that public opinion on all views has to be Elite Liberal (80-100) or close to it.
(2) happens if a 2/3 majority in both houses of Congress are elite liberal.
(3) happens if a 2/3 majority in both houses of Congress are arch-conservative.
All three need the usual passage by 3/4 of the state legislatures in order to pass. State legislature voting is determined solely by public opinion and doesn't care about laws.
Law changes
Laws are changed four times a year. These are:
1. Congressional legislation, which occurs on March 1
2. Supreme Court decisions, which happen on (I think) July 1 (give or take a month)
3. Congressional legislation AGAIN, on September 1
4. Initiative petitions, on November 1.
Elections for Congress and the President also happen on November 1 in appropriate years. See the actual U.S. Constitution for details (although in the actual U.S. Constitution, it's the Tuesday after the first Monday in November, not November 1).
(1) and (3) depend largely on the alignment of Congress and the President, although moderates will tend to be swayed by public opinion on issues.
(4) depends solely on public opinion.
(2) has nothing to do with public opinion; only the alignment of Supreme Court justices matters. However, there's a 40% chance of replacing a Supreme Court justice every year, which means that the President and Senate select a new one, compromising as they do so. The Supreme Court is also automatically more liberal than normal on Free Speech and Flag Burning laws, but more conservative on Gun Control.
What this means for gameplay is that if you are really, really effective at changing public opinion, the longest part of the game may consist of simply waiting for the government and the laws to catch up, which in Marathon mode can take several decades.