In a sentence, when is it proper to use "me" or "I"?

It was taught to me that if you say the sentence separately, it is easier to decide which sounds correct. Example:
"Sally and me went to the park".
"Sally went to the park".. correct.
"Me went to the park".. incorrect..
"I went to the park" .. correct
"Sally and I went to the park"... correct.

You could also use a preposition with it to say,
"Sally went WITH me to the park."
Hope this helps..

Use "I" when it's the subject. In other words, when you are doing the action.
*I went to the movies.
*I like this song.
*Bob and I are going to the store.

Use "me" pretty much any other time.
*Give that to me.
*Come to the store with Bob and me.
*You ran right past me.

Use "I" if you're the subject.

Use "me" if you're the object.

I threw the ball.
Bobby threw the ball to me.

If there are multiple people in the sentence, the easiest way to get it right is to break the sentence into individual ones for each person.

Sally and I walked to the movies.
Sally walked to the movies.
I walked to the movies.

My parents punished my sister and me.
My parents punished my sister.
My parents punished me.

All of the other answers are correct, but their is one other area when people miss use "me". That is in comparisons. People will often say something like, "He is [add descriptive term here] than me." This is incorrect. It should be, "...than I." The way to better remember this is to remember that you can complete the sentence by adding "am" to the end. So, it would become, "...than I am." "...than me am." doesn't sound right and is incorrect. You don't need "am" at the end of such a sentence, but putting it in, at least mentally, helps to keep straight which pronoun to use.

Examples:
"He is hungrier than me." (incorrect)
"He is hungrier than I." (correct)
"He is hungrier than I am." (correct)

The answers above are correct... but there are exceptions. "Who's going with us?" You would either say "I am," or "Me." The reason was explained by John McWhirter in one of his books on linguistics--that English has an oblique case for use in exclamations and such.

Don't say "For you and I," though. It's "For you and me" because it's "For you and for me."

julie plans to travel this summer with oscar and me.

My brother and I like to go fishing? Is the correct use of the pronoun I ?

Well I think the problem there is that neither phrase forms a complete sentence, so only once a verb is added can the your question be answered, For instance, both of the following would be correct.

Jack and I are in the wedding picture.

This is a picture of Jack and me.

It seems people comment pictures incorrecty all the time. Would you say " Jack and I at our wedding"?
That's what I see all the time, I think it should be "Jack and me at our wedding".
"Jack at our wedding" and "Me at our wedding".

Maybe I'm wrong...any thoughts?

I never imagined that me and you would ever bond as friends the way we did.

My brother and I like to go fishing? Is I correct in this case.

please feel free to contact me or Greg if you have any questions or comments.