Learn Hardcore Romanian: Eu mănânc micul dejun și după aceea plec la școală. - I eat breakfast and after that I leave for school.

Subject pronouns like Eu are optional in Romanian because the verb endings indicate person and number. You can omit Eu unless you want to emphasize the subject. So both Eu mănânc micul dejun and Mănânc micul dejun are correct.

What form is mănânc, and how is the verb a mânca conjugated in the present tense?

Mănânc is the 1st person singular present of a mânca (to eat). Present tense conjugation:
eu mănânc, tu mănânci, el/ea mănâncă, noi mâncăm, voi mâncați, ei/ele mănâncă.

Why is it micul dejun instead of mic dejun or dejun mic?

Romanian uses a postposed definite article. For masculine nouns, –ul attaches to the noun (dejun → dejunul). But when there's an adjective (mic), the article attaches to the adjective: mic → micul. So micul dejun literally means the small lunch, i.e. breakfast.

What does și după aceea mean, and can we use another expression?

Și means and, după aceea means after that. You can omit și and simply say după aceea, or use apoi (then) for brevity:
Mănânc micul dejun, apoi plec la școală.

Why do we say plec la școală instead of just plec școală?

The verb plec (from a pleca, to leave/go away) requires the preposition la when indicating a destination: la școală = to school. Without la, the phrase is ungrammatical.

How is plec conjugated in the present tense for other persons?

Present tense of a pleca:
eu plec, tu pleci, el/ea pleacă, noi plecăm, voi plecați, ei/ele pleacă.

How do you pronounce mănânc and școală?

ă is a mid-central vowel (like the ‘a’ in sofa).
ș is pronounced like English sh.
So mănânc sounds like muh-nunk and școală like shko-uh-luh.