Learn Portuguese (Portugal): Eu estudo trinta minutos.

The contractions à and às mean "at the" and are used to say exactly when something happens on the clock (for example, "at two o'clock" is às duas horas). This sentence is about how long an action takes, which is a duration. When stating a duration, you don't use à or às.

In English I might say "I study for thirty minutes." Do I need a word for "for" in Portuguese?

No, you don't need it here. Just like in English where you can simply say "I study thirty minutes", Portuguese directly connects the verb to the duration without a preposition. You just say estudo trinta minutos.

Does the number trinta change depending on whether the time word is masculine or feminine?

No! In Portuguese, only the numbers 1 and 2 (and the hundreds) change their endings to match masculine or feminine words. The number 30 is always trinta, so it stays exactly the same whether you are counting minutos (masculine) or horas (feminine).

Is it normal to include Eu at the beginning, or do people usually drop it?

While it is perfectly correct to include Eu, native Portuguese speakers drop it most of the time. The verb ending in -o (estudo) already clearly shows that "I" am the one studying, making the Eu optional in everyday conversation.