Is this correct? “24th and 25th August for my mom and I” or “24th and 25th for my mom and me”?

The correct phrasing would be "24th and 25th for my mom and me." The use of "me" is appropriate here because it is the object of the preposition "for." If you were using the phrase in a different context where "I" would be the subject, you would say "my mom and I are going." In this case, though, since it's part of a prepositional phrase, "me" is the correct choice.