''Me and my dog'' or ''my dog and I''

Which one is correct?

This depends:

Say the sentence, but omit "my dog"....for example:

"My dog and I went to the park."

"I went to the park."

It wouldn't make sense to say "Me went to the park."......so "I" is the correct choice.

Now take the sentence:

"The man at the pet store gave my dog and me dirty looks."

"The man at the pet store gave me dirty looks."

It wouldn't make sense to say "he gave I dirty looks"......so "me" is the correct choice.

Mjd's answer is correct with respect to where to use "I" or "me" but there is more to the question. In what order do you put them? Which of these sentences would you say?

A1a) "My dog and I went to the park."

A1b) "I and my dog went to the park."

And what about these?

A2a) "The man at the pet store gave my dog and me dirty looks."

A2b) "The man at the pet store gave me and my dog dirty looks."

It is normal to put yourself after other people. This is a matter of courtesy but would you extend that courtesy to a dog? I wouldn't, I'd rather go with the b) sentences.

What if it weren't a dog but an inanimate object like a computer? Which of these sentences would you use?

B1a) "My computer and I were at the park."

B1b) "I and my computer were at the park."

And what about these?

B2a) "The man at the electronics store gave my computer and me dirty looks."

B2b) "The man at the electronics store gave me and my computer dirty looks."

Again I'd go with the b) sentences, wouldn't everyone?

I really want to go back to my primary school and yell at my English teacher, "Where you put 'I' in a list of subjects isn't a matter of grammar! If I have absolutely no respect for my fellow subject, I reserve the entirely grammatical right to put myself before him in a sentence!"

Hear! Hear!