The second sentence is the more correct one. The grammatically correct pronoun is 'I', because this is part of the subject of the sentence (nominative case).
However, I'd be reluctant to say that the first sentence is actually 'wrong'. Why? Because hundreds of millions of native speakers use 'me and ..' in this way. I use it myself in informal conversational contexts. Strictly speaking, it doesn't follow the rules of formal grammar, because 'me' is an object pronoun. It's OK to say this informally, but not when you're taking a grammar test or writing a formal text.