Learn Hardcore Romanian: Începutul este bun.

Romanian places its definite article as a suffix rather than a separate word. For most masculine singular nouns, you add -ul (or -le after certain consonants). So început (“beginning”) + -ul = începutul (“the beginning”).

What is the gender and case of începutul in this sentence?

Începutul is a masculine, singular noun in the nominative case. It functions as the subject of the sentence. In Romanian, nouns have gender (masculine, feminine or neuter) and show case mostly by context or by adding articles and adjective endings.

Why is the adjective bun used instead of bine?

Bun is an adjective (“good”) agreeing with a noun; bine is an adverb (“well”). Since you’re describing începutul (a noun), you need the adjective form bun.

Could you say Începutul bun and drop the verb este?

No. In standard Romanian you need the linking verb este (“is”) to form a complete sentence: Începutul este bun. Omitting este would just leave two words (“beginning good”) without a verb.

Can I use the shorter form e instead of este?

Yes. In colloquial speech and informal writing, e replaces este:
Începutul e bun.

Is it OK to change the word order to Bun este începutul?

Yes. Romanian word order is relatively flexible. Starting with Bun puts extra emphasis on the adjective and can sound more poetic or formal:
Bun este începutul.

How would you say “a good start” instead of “the beginning is good”?

Use the indefinite article un and put the adjective after the noun:
Un început bun.

How do you pronounce Începutul este bun?

Phonetically:
/ɨn.t͡ʃeˈput.ul ˈes.te bun/

î is a close central unrounded vowel [ɨ],
ce = [t͡ʃe],
-ul at the end = [ul].