There are rigorous rules that can be applied; however these can sometimes be confusing. The simplest and easiest way to intuitively check is to simply remove the other pronoun and then see whether the phrase makes sense and sounds right.
From the above, it's clear that "Jack is going to the fair with I" is incorrect.
As another example, should it be "Has John seen Andy and I?" or "Has John seen Andy and me"? Again we take out the other pronoun. Clearly "Has John seen I?" is incorrect.
And finally, is it "Henry and I are well" or "Henry and me are well"? Again we take out the other pronoun and in this case we change the plural "are" to "am". Clearly "Me am well" is incorrect. It should be "I am well".
• By Etienne de L'Amour ~ Google+