To some people’s ears, the sentence: “Terry is going shopping with June and I” sounds perfectly reasonable and correct. To those same people, the statement: “Terry is going shopping with June and me” might well sound rather colloquial and improper.
In fact, the latter version is grammatically correct. A simple set of guidelines helps explain.
Firstly, as in both cases above, whenever you are listing people including yourself, you should always deferentially name yourself last (e.g. “Terry and me” rather than “me and Terry”).
The greater problem is when to use “me” and when to use “I”. Basically, if you are part of the subject of the sentence (which is ordinarily indicated by appearing near the start of the sentence), then you need to use “I”. If you are not part of the subject of the sentence but one of the objects (ordinarily indicated therefore by appearing later in the statement), you should refer to “me”.
So, correctly, you could say: “Terry and I are going shopping with June” or “Terry is going shopping with June and me”.
You should not say: “Terry and me are going shopping with June” or “June is going shopping with Terry and I”.
Confused? Not sure? Try the above incorrect sentences with Terry staying at home. You basically end up with: “Me am going shopping with June” or “June is going shopping with I”. They obviously sound wrong, so you know you need to try again.