Is it correct to say me and dad went to the football game?

No: it is never correct to use "me" as a subject. The correct form is "Dad and I went to the football game." An easy test for whether you should use "me" or "I" is to try the sentence with "me" or "I" alone, so that you have a simple rather than compound form.

It is easy to see that "I went to the football game" is correct, and "Me went to the football game" is not correct.

Note that if you are using "me and Dad" as a compound object, that is perfectly correct. For example, it is correct to say, "This means so much to me and Dad" and incorrect to say, "This means so much to Dad and I." You can use the same test here: it is easy to see that "This means so much to me" is correct, and "This means so much to I" is not correct.