Can I access the computer's libraries?

Hint 1 of 7:

Of course.

Hint 2 of 7:

You should first probably have Spock examine the large sphere with his
tricorder to see what he can make out of it.

Hint 3 of 7:

But before you can decode the data in the spheres, you'll need to know a
bit about the alien's culture.

Hint 4 of 7:

That information can be found on the bridge.

Hint 5 of 7:

Have Spock use the left chair on the bridge to learn about the alien's
fixation on certain numbers (especially 3).

Hint 6 of 7:

Then Spock can use his tricorder on the sphere again.  McCoy will offer
his tricorder as well, and you'll download the information.