Today we're going to learn a lesson about pronouns from the delightful, inspiring, and just a tad grammatically incorrect Olympics theme song.
Does the Olympic Theme Song, “I Believe,” Have a Grammar Error?
I believe in the power that comes
From a world brought together as one
I believe together we'll fly
I believe in the power of you and I
Lest you think I'm being a snobby American, I want you to know that I've been getting e-mail messages from Canadians ever since the song debuted. Here's one from Gilda in Thornhill, Ontario:
As a Canadian, I’m very proud and excited that my country will be hosting the upcoming Winter Olympics in Vancouver. Our national broadcaster of the Games has started to play their official theme song, “I Believe,” and it really is gorgeous, inspiring, and uplifting.
However, as much as I love it from a patriotic and sports-lover point of view, I have a problem with it. One of the lines in the chorus, repeated in several places in the song, is “I believe in the power of you and I” (to rhyme with “fly"). 
I realize they needed “I” instead of “me” for the rhyme, but it still offends me.
As an Olympic fanatic, I know I am going to hear this song a million times in the next three weeks, but as a grammar stickler, it’s going to bug me EVERY SINGLE TIME I hear it. What do you think?
Well, Gilda, what do I think? I think we shouldn't be surprised when songs are grammatically incorrect because they're like poetry, and there's a reason breaking the rules is called taking poetic license.
But I share your cringing and hate to think that millions of people are singing along absorbing the incorrect phrase into their subconscious so that the next time they have to write about the power, the myth, the clash, or the best you and me, they instead write about the power, the myth, the clash, or the best of you and I.
So let's do what we can to set people straight and talk about pronouns.
What Are Pronouns?
Pronouns fill in for nouns. I think of them as stuntmen and women, stepping in for nouns when the going gets tough, or just when nouns get tired. Instead of saying something like “Kenny went to Kenny's car,” you replace the second instance of “Kenny” with a pronoun so we don't have to hear Kenny's name over and over: Kenny went to his car. “His” is the pronoun that replaces the noun “Kenny.”
Just so you know the terms, “Kenny”--the noun the pronoun is filling in for—is called the pronoun's antecedent.
Now, because they don't get the recognition they deserve, some pronouns can be temperamental. Some will only work when they can be the subject of the sentence, and others will only work when they can be the object of the sentence.
“I” is exclusively a subject pronoun, and “me” is exclusively an object pronoun.
What Is the Difference Between a Subject and an Object?
Subjects are the ones taking action in a sentence, and objects are the ones receiving the action.
In the sentence “Squiggly slimed Aardvark,” “Squiggly” is the subject because he's the one doing something, taking the action, sliming. “Aardvark” is the object because he's the target of the action. He's not actively doing anything; he's just getting slimed.
 “I” Versus “Me”
When a pronoun steps in for a subject noun, it has to be in the subject form.
I slimed Aardvark.
I hugged Squiggly.
I thanked Kenny.
In each of those sentence, you can tell “I” is the subject because it refers to the person taking an action: sliming, hugging, and thanking.
When a pronoun steps in for an object noun, it has to be in the object form.
Aardvark slimed me.
Squiggly hugged me.
Kenny thanked me.
In each of those sentences, you can tell “me” is the object because if refers to the person who is the target of the action: the one who is being slimed, hugged, and thanked.
Prepositions and Objects
You may be thinking, in the lyrics “I believe in the power of you and I,” (cringe) it doesn't seem as if the pronouns--“you” and “I”--fit the model of that simple sentence you gave us with a subject, verb, and object. And you're right.
The clue in “I believe in the power of you and I” (cringe) is that the pronouns follow the word “of.” They're part of a prepositional phrase, and when pronouns follow prepositions, they are always in the object case. It's just one of those things you have to know—the only pronouns you'll find following prepositions are object pronouns.
It can help though to notice that those pronouns aren't taking an action, so it's pretty clear they shouldn't be subject pronouns.
The cloud was above him.
Water rained over me.
The river ran through it.
The secret was between them.
In each of those sentences, the pronoun following the preposition wasn't referring to a person or thing doing anything. The pronoun wasn't taking any action, so you could tell you shouldn't use a subject pronoun because only subjects take action.
I Believe in You and I or You and Me?
So now, which one is the object pronoun: “I” or “me”?
“Me!”
So which pronouns should follow “I believe in the power of”?
“You and me!”
So when you hear the song “I Believe,” be inspired, be uplifted, but don't be snookered into messing up your grammar. Remember that the correct way to say that line is “I believe in the power of you and me.”