As I told you in class, this is a wonderful book because it covers many aspects of the English grammar. Here you are, for example, the answer to the title of this post.
1- The English rule is to use ‘I’ for the subject of a sentence (that means the person who does the action) and ‘Me’ for the object of the sentence (that means the person the verb is acting upon). Thus, it is not correct to say ‘Me met Mary’ but ‘I met Mary’ or ‘Peter saw I‘ but ‘Peter saw me’
This rule is exactly the same in compound subjects or compound objects, that is, a subject or an object that consists of more than one noun or pronoun. So, ‘Peter and I went to the cinema’ is correct, because Peter and I are the subject, the ones performing the action. And ‘Martha has invited my husband and me for dinner’ is correct, as my husband and me are the object, the persons receiving the action of the verb, the invitation.
2- And what happens with the verb ‘to be’? What would you say here: ‘It is I’ or ‘It is me’? ‘It wasn’t I who said it’ or ‘It wasn’t me who said it’?
Traditionally, the first option should be the correct one, because following the Latin rules, the complement found after the verb ‘to be’ has a subject function. However, nowadays the second option is becoming acceptable and the first option is becoming old-fashioned.
In comparatives, these structures are accepted: ‘He isn’t as fast as me (or I)’ and ‘He isn’t as fast as I am’
3- Remember that after prepositions, the rule is always to use an object pronoun, thus: ‘between you and me’ (not I) or ‘she never meant anything to me’ (not to I).
4- The book also highlights the fact that these rules do not apply to songwriters. So, don’t try to learn grammar from lyrics, as you will commonly find expressions like: ‘Me and Mrs Jones, we’ve got a thing going on…’
(explanations adapted from ‘My Grammar and I (or should that be ‘Me’?)’ by Caroline Taggart and J.A. Wines)