Usage of "I" and "Me"

Distinguishing when to use "I" and when to use "me" is sometimes confusing when connecting them with another pronoun in a sentence. People assume that whenever connecting the two pronouns with and, you're supposed to use "and I."  For the longest, that is what I thought.

Common Phrases Heard:

She told Jane and I to walk to the store.
Gary bought basketball tickets for him and I.

I read a neat tip on knowing when to use what word.  When you omit the pronoun and conjunction word from the sentence, does it make sense? For example, when correcting the two sentences above, does this make sense?

She told Jane and I to walk to the store. Correct Way:  She told Jane and me to walk to the store.

Gary bought basketball tickets for him and I. Correct Way: Gary bought basketball tickets for him and me.