Does the first amendment protect your right to see whatever movie you wish?

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The first amendment protects the right to express yourself, including:
Free speech
Free exercise of religion
Freedom of the press
Peaceful assembly
Right to petition

I'm assuming you are asking about children getting into R rated movies? No, this is not a right.

No, movies are restricted by age

And of course movie theater are private owned and they can restrict who they serve

The "state " must demonstrate that a limitation on freedom of speech and expression is necessary ;otherwise the 1st amendment is absolute. This has left it up to the court system and the legislative branchs to define the parameters and defend their decision in the judicial system. This is logical in that these freedoms are sometimes in conflict with other values.
Regarding movies then ,the court defined in 1973 the "Miller Test " (Miller v California) that "obscenity " was not a right protected under the 1st amendment.
The Miller test says that a movie is not protected if the average person, applying contemporary community standards, would find that the work, taken as a whole, appeals to the "prurient" interest.
Whether the work depicts/describes, in a patently offensive way, sexual conduct or excretory functions specifically defined by applicable state law,
Whether the work, taken as a whole, lacks serious literary, artistic, political or scientific value.

The rating system mentioned is a voluntary system that the motion picture industry agreed to.

does the first amendment protect your right to see whatever movie you wish?

In a word; No!

The First Amendment restricts what laws the government can make limiting one's right to say what they want. The movie industry is NOT the government, Therefore, it has every right to restrict access to their content based on age or other criteria.