Which is correct? Tim and I got sick. or Tim and me got sick.
You must use a subject pronoun (I, we, you, he, she, it, or they) as the subject of a sentence.
Tom and I got sick.
He and I got sick.
We got sick.
Me, us, you (yes, you is the same in both cases) him, her, it (same again) and them are the object pronouns.
Tom and I gave him and her our sickness.
They gave it to them and us.
We gave it to you.
You gave it to me.
The subject does the action, the object receives the action. It gets much more complicated than this, but this is a good place to start.