I, you, he, she, it, we, they = subject pronouns
Me, you, him, her, it, us, them = object pronouns
So 'My sister and I are alike' is the correct form. 'I' is part of the subject of the sentence.
'My sister and me are alike' is wrong, because you should not use 'me' for the subject of a sentence.
'Me' is used for the object of the sentence, for example:
'Dad took a photo of my sister and me.'
In this case, 'me' is the correct pronoun and 'I' would be wrong. 'Dad' is the subject of this sentence, and 'my sister and me' is the object.
NB A shameful number of native English speakers get this last point wrong. Many English speakers mistakenly believe that 'me' is casual and 'I' is formal, regardless of the grammar of the sentence.