Because if Rita wasn’t in the picture… you would say “I am going to the beach…”… so you use “I” not, me…. I know that the “am” changes to “are”… but that is because two people as the subject requires a plural “verb”….

It’s not “me”… because you wouldn’t say “me am going to the beach”…. it just sounds wrong because it IS wrong…

And you wouldn’t list yourself first…. you would list yourself last… that’s why “b” is wrong… besides the fact that the usage of “me” is wrong…

You use in phrases like “between you and me”…. or “he threw the book at Johnny and me”….

“Rita is going to the beach and I am going to the beach”.

Now try that construct with (b) and (c).

“Me is going to the beach and Rita is going to the beach”

“Rita is going to the beach and me is going to the beach”

are both obviously wrong.

Objective case pronouns (me, you, him, her, it, us, and them) are used as direct objects, indirect objects, and objects of prepositions, not subjects.

Hope this helps!