Learn Hardcore Serbian: Biram ribu sa spiska. - I am choosing fish from the list.

Serbian dictionaries usually give the infinitive form, so learners often first see birati, but in an actual sentence you usually need a conjugated form like biram.

Where is the word for I?

It is understood from the verb form.

In Serbian, subject pronouns are often omitted when they are clear from the verb:

biram already means I choose
so ja biram is possible, but ja is often unnecessary

You would add ja only if you want emphasis or contrast:

Ja biram ribu, a on bira meso.
I’m choosing fish, and he’s choosing meat.

Why does riba become ribu?

Because it is in the accusative case, which is used for the direct object of the verb.

Here, the thing being chosen is fish, so riba changes to ribu.

For many feminine nouns ending in -a:

nominative singular: