Learn French: Je crois que la lune est magique. - I believe that the moon is magical.

Croire (to believe) is an irregular verb. In the present tense, the I form is je crois. The you (singular) form is also tu crois, and the he/she/it form is il/elle croit (with a t instead of an s).

In English I can say "I believe the moon is magical." Can I drop que in French?

No, you can never drop que (that) in French when connecting two parts of a sentence like this. Even though it is optional in English, you must always say "I believe that..." (Je crois que...) in French.

La lune is feminine, so shouldn't the adjective magique have an extra e?

When an adjective's default masculine form already ends in a silent e (like magique), it doesn't change for the feminine. It stays exactly the same: un moment magique (masculine) and la lune magique (feminine).

Can je crois que also mean "I think that"?

Yes! While it literally means "I believe that," French speakers frequently use je crois que in casual conversation to mean "I think that," very similarly to the verb penser (to think).