There are grammatical reasons for why you use "I" rather than "me" in a sentence, but I just want to simplify things. Use "I" when the next word is a verb, use "me" when the next word is not a verb.
Examples:
✓ I am going for a coffee.
✗ Me am going for a coffee.
✓ Rose and I went for a coffee.
✗ Rose and me went for a coffee.
✓ The dog followed me.
✗ The dog followed I.
✓ Rose spent the day with me.
✗ Rose spent the day with I.
NOTE: Don't be afraid to use "me".
✓ Rose spent the day with Rachel, Trish, and me (not I).
In most cases, "I" is the subject of the sentence: the person, place, thing, or idea that is doing or being something. In this example, Rose is the subject, not me.
BUT...
Rose and I spent the day with Rachel and Trish (because I am now the subject).
There are always exceptions when you can end a sentence with "I" (dang English language). For example, when a negative, emphatic, or certain adverb occurs at the initial position of a sentence, the subject and the verb get inverted. An inversion occurs in the interrogative as well. Some examples follow:
Neither was I.
So was I.
There was I.
Where was I?