Julie and me walked in the park today.
This is very common usage, but ‘me’ is the wrong personal pronoun. Remove ‘Julie and’ from the sentence above and you’re left with:
Me walked in the park today.
Which is (if you’re a native English speaker other than Ralph from The Simpsons) clearly incorrect! The correct phrase is:
Julie and I walked in the park today.
‘Julie and I’ is the subject of the sentence; remove ‘Julie and’, making ‘I’ the subject, and it still makes sense. Removing the other nouns from the sentence is a useful way to check whether you’re using the right pronoun.
It’s common for people to get the reverse usage wrong too, as in the following example (adapted from the Oxford English Dictionary):
Rose spent the day with Jake and I.
This should be:
Rose spent the day with Jake and me.
Here ‘Jake and me’ is the object of the sentence. Remove ‘Jake and’ and it still makes sense.