The correct definition for this table is as follows:
CREATE TABLE user_movies (
user_id INT NOT NULL,
movie_id INT NOT NULL,
PRIMARY KEY (user_id, movie_id),
FOREIGN KEY (user_id) REFERENCES users(user_id),
FOREIGN KEY (movie_id) REFERENCES movies(movie_id)
) ENGINE=InnoDb;
Notice "primary key" is a constraint, not a column. It's best practice to have a primary key constraint in every table. Do not confuse primary key constraint with an auto-generated pseudokey column.
In MySQL, declaring a foreign key or a primary key implicitly creates an index. Yes, these are beneficial.