In English we have lost most of the uses for cases, e.g., the Nominative, Accusative, Dative, Genitive, Ablative or Vocative that are needed in languages such as German, Russian or Latin.
Nevertheless, they are retained in certain situations and one of those is with pronouns, e.g., I, me or my.
Although you may not know the grammarian rules behind it, as a native speaker you will instinctively know which case of each pronoun to use in most situations.
You would, I hope, know that 'I play tennis' rather than 'Me play tennis' is correct.
The confusion arises when there is a named subject or object along with the pronoun. A person's name doesn't need a case in English (except with possession in the genitive case, where it takes an apostrophe), so it is able to be inserted in most places unchanged. Since 'Clare went to the cinema' makes perfect sense, shouldn't 'Me and Clare went to the cinema' make sense too? No, since as rdizzle points out, 'Me went to the cinema' makes no sense on its own and the clause is ungrammatical. The correct form of each pronoun needs to be chosen, whether it is I, me or my, he, him or his, she, her or her and so on.
Finally, don't fall into the trap of hyper-correction and inserting 'I' (the nominative) all the time thinking it is correct. 'Would you like to come to the cinema with Clare and I' is wrong, since you would not say 'Would you like to come to the cinema with I?' Here the accusative (me, the receiver of the verb or the object) rather than the nominative (I, the doer of the verb or the subject) is appropriate.
Hope that helps.