Learn Polish: Poznałem twoją siostrę wczoraj.

In Polish, the verb ending already tells us who is doing the action. The -łem ending on poznałem specifically means 'I' (a male) did it in the past. Adding Ja is grammatically correct but usually left out unless you want to emphasize that I was the one who met her, not someone else. If a female speaker were saying this, she would use poznałam.

Why do the words twoja siostra change to twoją siostrę here?

They change because they are the direct object of the sentence - the person receiving the action of being met. This requires the Accusative case. For feminine words ending in -a like siostra, the noun ending changes to -ę (siostrę). Adjectives and pronouns describing that noun, like the possessive twoja, change their -a ending to -ą (twoją).

Could I use the verb spotkałem instead of poznałem to mean 'I met'?

It depends on the context! You use poznałem (from the verb poznać) when you mean 'met for the first time' or 'got to know' someone. If you already knew her and just ran into her or met up with her yesterday, you would use spotkałem (from the verb spotkać).

Is it okay to put wczoraj at the beginning of the sentence?