“We” is a subject form, “us” an object. We do things; things are done to or for us.
If this doesn’t help, you can try a couple of simple tests. If you are clear about the difference between “I” and “me,” try making your sentence singular. “We” becomes “I” in the singular and “us” becomes “me.”
“Our mothers and us are going shopping” becomes “my mother and me are going shopping”—which is wrong. So the sentences should read “My mother and I are going shopping” and “Our mothers and we are going shopping.”
But if that doesn’t seem obvious, try eliminating everything but the pronoun and the verb: “Us are going shopping” should be “we are going shopping.”
Test a sentence like “us girls have sold more calendars than the guys” by reducing it to “us have sold.” This sounds wrong. It should be “We girls have sold.”
But “they gave us girls the prize” is correct because “they gave us the prize” is also correct.