Have a look at these questions:
a) "Does the cab take us at home?"
b) "Does the cab pick us (up?) at home?"
Both questions mean the same? Is there any difference between "to pick (up?)" and "to take" (in these questions)? Do I have to use "up"?
"Does the cab take us home?" This makes sense since you are wondering if this taxi is going to your home. "Take" in this context means "deliver to". It does not mean to "pick up" or "collect"
"To pick up" can mean several things depending on context.
Pick up your clothes off the floor. Don't be so messy! Here the meaning is literal. You want the clothes literally removed from the floor and put away somewhere.
I'll pick you up at 7:00pm from your home and we can go to dinner. Here "to pick up" means that I will come to your house usually in a car at 7:00pm and we will then go out to dinner.
So this question - "Does the cab pick us up at home?" - is asking whether the cab will collect you at your home or perhaps this is not convenient and another place is better.
Question: "Does the cab pick us up at home?"
Answer: "No, I told the driver to go to Mike's house. We'll catch it from there."
Hmmm..., yep. I did understand it.