Learn Italian: Credo che la sagra sarà un grande successo. - I believe that the local festival will be a big success.

This is exactly what the current lesson covers! While you use a before a noun or pronoun (like credo a questa storia for "I believe this story"), Italian strictly forbids using prepositions directly before the declarative conjunction che (that). Therefore, the preposition is mandatorily dropped, leaving just credo che.

Doesn't the verb credere trigger the subjunctive? Why are we using the future indicative sarà?

You are completely right that verbs of opinion like credere normally require the subjunctive mood. However, Italian does not have a future tense in the subjunctive. To express an action that will happen in the future ("will be"), you drop the subjunctive and simply use the regular future indicative (sarà).

What makes a sagra different from a festa or a festival?

A sagra is a specific type of local, traditional outdoor festival in Italy, almost always centered around celebrating a regional food product (like truffles, wine, or chestnuts) or a patron saint. A festa is a more general word for any party or holiday, while a festival usually refers to a larger cultural or arts event (like a film or music festival).

Why does grande come before the noun successo instead of after it?