personal pronouns - Usage of "Me and yours"

I was having a chat to my girlfriend when something came up and I said

"Speaking of dads, I had a dream where me and yours were going...".
Here "me" refers to myself and "yours" to her dad.

She corrected me saying that this was incorrect, however could not quite identify where. Later I said that I should've replaced "me" with "I" and swapped the order around. However, this still felt clumsy and she insisted that was still incorrect.

What is a good, conversational, way that this might be said? Furthermore, are there any correct uses of "me and yours" , idiomatic or otherwise?

Answer

I had a dream where me and your's were going...

It still feels clumsy, because your omission of "dad" adds unnecessary implications.

Let's simplify first, then add onto it again:

I had a dream where I was going...
I had a dream where me was going...

Clearly, the first one is correct. You need to use "I", not "me" because it is the subject of the second clause.

But the further confusion stems from "me and yours". You mean that you and her dad are the subjects, but because you omit the "dad" from "yours", you imply that it was also your dad instead of you.

"My dad and your dad." can change into "My dad and yours". The second "dad" can be omitted because the first "dad" reference is already there. Without that prior reference, omitting "dad" only leads to unintended implications.

It seems best to avoid that implication as it only makes an already loaded phrasing much more prone to be misunderstood.

And since we're reworking anyway, let's flip it to "your dad and I" because of politeness.

I had a dream where your dad and I were going...

Edit

I had a dream where me and your dad were going...

Would be the way I expect to hear it when spoken, but isn't grammatically correct. As you can see from this example though, even by only removing the "dad" omission, you solve most of the confusing language problem.

Edit 2

If you are Irish, the possessive "my" is often pronounced as "me". In case you are Irish, I would interpret the following:

Speaking of dads, I had a dream where me and yours were going...

as the Irish equivalent of

Speaking of dads, I had a dream where [mine] and [yours] were going...

Which would also be correct (that is to say, accepted informally), assuming it's about both your dads, not you and her dad.