SUGGESTION (приглашение что-то сделать)
shall – should – could – let’s – why don’t – have

It’s a lovely day. Shall we go for a walk? (… (Может) сходим прогуляемся?) ~ Yes, OK.
Let’s play some music. ~ Good idea.
We could watch this comedy on TV tonight (… могли бы посмотреть эту комедию…) ~ Well, actually, I’ve seen it before.
You could invite a few friends around(Ты мог бы пригласить нескольких друзей…) ~ Yes, why not?
Shall we go to the park? ~ Good idea (Может сходим в парк?)
Why don’t…?
Why don’t we have a look round the market? (Почему бы нам не пройтись по базару?)

OFFER (предложение (сделать что-то ДЛЯ КОГО-ТО)

1. ГОТОВНОСТЬ
will – can  
I‘ll carry your bag. ~ Oh, thanks.
We can give you a lift. ~ Oh, that would be great.

2. ВОПРОС
shall – should – can – could
Shall I wait for you? ~ Oh, thanks. I won’t be long ((Мне) вас подождать?)
Shall we pay you the money now? (Вам сейчас заплатить?) ~ Oh, there’s no hurry.
Who shall / should I pass the message to?

Shall / should I phone for a taxi for you? (Может вызвать вам такси?)
Should I put these books over here?
Can I get a taxi for you? (Могу я вызвать вам такси?) ~ Yes, please.
Could I help you with your bags? (Давайте помогу вам с сумками)

СПРОСИТЬ СОВЕТ, рекомендацию
shall – should

I’m in terrible trouble. What shall / should I do? (… Что мне (следует) делать)
Where shall / should we go for our holiday? (Куда поедем / куда бы поехать…? ) ~ What about Spain?
What shall / should we cook for the birthday party?

ПРИГЛАШЕНИЕ
Would you like to …?
Would you like to visit the museum with us?
Would you like to have lunch with us? ~ Yes, I’d love to. Thank you.
Would you like to stay the night? ~ Oh, that’s very kind of you.
Would you like me to get you some water? (NOT Will you like me to get…?)

Will / Won’t you …?
Will you join us for coffee? ~ Yes. Thanks.
Won’t you sit down?

В неформальной речи – императив:
Come and have coffee with us.
Have a biscuit. ~ Thank you.