I'm not sure if you're excluding the Fourteenth Amendment as an extremely important and necessary amendment here or not, but I'll go ahead: the Fourteenth Amendment guarantees citizenship, and the rights that come with it, to everyone born in the United States. Even with the Thirteenth Amendment abolishing slavery, prior behavior by states - North and South - and the federal government had basically established that even free blacks weren't citizens and had none of the rights that whites held. After the Civil War, this discrepancy allowed the South to enforce Black Codes, which enabled a de facto form of slavery - unemployed blacks could be arrested and "indentured" to whites. The Fourteenth granted all of the Constitution's rights to all Americans regardless of race, making this injustice untenable. Additionally, it in effect made it so that the states had to follow the Bill of Rights as well as the federal government.
I'd have to agree that the First Amendment is the most important amendment. All the other rights are also necessary, don't get me wrong, but the First implicitly establishes freedom of thought. All governments which have restricted freedom of expression and religion have in reality attempted to control the minds of men and force them to become subservient mentally as well as physically. The First Amendment sets man's mind a a sacred object, and rightly so - all human progress and liberty requires freedom of thought.