I and me are often used incorrectly by second language learners. A different word is used depending on whether or not the person is the subject or object of the sentence. The subject is the thing doing the action, the object receives the action.
This language problem is leftover from ancient English (Anglo Saxon). Long ago English words were all this way, with different words used depending on whether or not the thing is the subject or object of the sentence. It would have been much more difficult to learn the original barbaric English. In ancient history as people learned English as a second language they simplified it. The result is modern English.
Use I, along with words (other pronouns) such as he, she, you, we and they, when the word is doing the action (is the subject) in the sentence.
I am late.
She went to school.
They waited for the MTR.
Tom and I are going to eat dim sum.
In the last example I and the name Tom, both do the action. So in these types of sentences you need to use I rather than me.
Use me, along with words (other pronouns) such as us, him, her, you, and them, when the word is receiving the action (is the object) in the sentence.
Jane kicked you.
Tom drove them to the park.
Mary thanked him for helping her study.
The taxi drove Mark and me to the wrong place.
In the last example, the word me, use together with the name Mark, both receive the action. So in these types of sentences you need to use me rather than I.
People often have trouble when the sentence uses "Mark and me" or "Tom and I". You can make understanding easier if you imagine the sentence with only I or me. Then compare it to the examples.
(Remember that John and I=we, John and me=us)
The word you is the only one used both as the subject and object. Some strange things happened to this word and we will cover that next week.