Exactly what CreatureKindness said.
So, when in doubt, there are two ways you can figure it out. Either...
a) Remove the other person from the sentence, eg "Dad took Liam and me/I to the movies" becomes "Dad took me/I to the movies". "Me" becomes the obvious choice. Or;
b) Switch the two people around, eg "Dad took me/I and Liam to the movies". You would never say "Dad took I and Liam to the movies", and it's just as incorrect to say "Dad took Liam and I to the movies".
As an aside, in this example putting the word "Liam" in front of the word "me" is not a grammatical rule, it's just polite. So, grammatically, it's no different to say "Would you like to come with I and Sam to the shops" as it is to say "Would you like to come with Sam and I to the shops"; both are equally incorrect. Yet this is how so many people are taught to speak.
ETA: BTW Pegasus, it never changed, your earlier teachers just had it wrong (as I would say the majority of them do).