To ask for help, Can you and Could you come before a verb, and Would you mind comes before an -ing word. Reach for Could you with people you don't know well, and Can you with friends.
Asking for help
Can you — Can you help me with this bag?
Could you — Could you open the door?
Would you mind — Would you mind holding this?
Try it: Mei: ___ help me with this heavy bag? Lars: Of course. Let me take it. Mei needs help and wants to ask Lars politely. Which fits? (Could you / Let me / Shall I) Answer: Could you — Could you is a polite way to ask someone for help, which is what Mei needs. Let me and Shall I are used to offer your own help, so neither one asks Lars to do something.
Offering help
To offer help, use Let me before a verb when you just step in and do it, Shall I to check first, and Would you like me to for a warm, careful offer.
Let me — Let me carry that for you.
Shall I — Shall I call a taxi?
Would you like me to — Would you like me to help?
Try it: Camila: I can't reach the top shelf. Farid: ___ get it for you. Choose the offer that means Farid will do it himself. (Let me / Could you / Would you mind) Answer: Let me — Let me means Farid will do it himself, which fits. Could you and Would you mind both ask Camila to do something, so neither one offers his own help.