Sam and I went to watch the ballgame together. Which kind of pronoun case is the bold word?

The correct answer is Nominative case.

This is the primary case where pronouns are I, You, he/she/it, we/you/they. In an objective case, the pronouns would be changed into me, him, her, them, us, and similar. Possessive would be if the pronouns became possessive pronouns and would be something like mine, yours, ours, and other similar forms of pronouns.