Article IV
Section 1. Full faith and credit shall be given in each state to the public acts, records, and judicial proceedings of every other state. And the Congress may by general laws prescribe the manner in which such acts, records, and proceedings shall be proved, and the effect thereof.
This is why you can get married in Nevada and it will be recognized in your home state.
That full faith and credit shall be given does not, in itself, mean that the states are required to alter their laws to accomodate the documentation from other states. It only means that, consistently with their own laws, the states must treat documentation from other states the same way it treats its own documentation. Hence, a marriage license from Hawaii between two men is only good in California to the extent that its laws allow men to marry each other.
If certain restrictions on marriage (such as age restrictions) can be evaded by going to Nevada or Utah or someplace like that, that's only because either the other states have agreed to allow it, or because Congress has mandated it, pursuant to the second sentence of the Section that you quoted.
I'm pinging our resident legal expert, just in case I'm mistaken about this, in which case I'll just have to hit the books again.