Learn Spanish (Latin America): Yo soy estudiante.

You can just say soy estudiante. In Spanish, the verb soy already means "I am," so the pronoun yo (I) is optional. It is usually dropped unless you want to add emphasis or avoid confusion.

Why isn't there a word for "a" like in "I am a student"?

In English, we use "a" or "an" with professions and roles, but Spanish skips this! When using the verb ser (to be) to state your job or that you are a student, you leave out words like un or una. You literally say "I am student."

Does the word estudiante change if a woman is speaking?

No, estudiante stays exactly the same whether you identify as male or female. Many Spanish words ending in -e or -ante are gender-neutral, so you don't need to change the ending.

Why do we use soy here instead of another form?

Soy means "I am" and comes from the verb ser, which is used to talk about identity, origin, and permanent traits. Since being a student is considered part of your identity here, soy is the perfect fit.