Learn French: Je suis tombé hier.

You are right, literally it translates to 'I am fallen.' While most French verbs use avoir (to have) to form the past tense, a specific group of verbs about motion or changing state use être (to be) instead. Tomber (to fall) is one of these motion verbs, so we must use je suis instead of j'ai.

If a woman is speaking, does tombé change?

Yes! A key rule when using être for the past tense is that the action word (the past participle) must match the subject's gender. If a female is speaking, she adds an e to make it feminine: Je suis tombée. Don't worry about pronunciation though, because tombé and tombée sound exactly the same!

How do we make the past form tombé from the dictionary verb tomber (to fall)?

Tomber is a regular -er verb. To make its past form (called a past participle), you simply drop the -er at the end and replace it with -é.

Can I put hier at the beginning of the sentence?

Absolutely. Just like in English, you can say Hier, je suis tombé (Yesterday, I fell). Time words usually go at the very beginning or the very end of the sentence.