Shall and Will are modal verbs primarily used to express the future.
In general or colloquial usage, will is used for all pesons, both sg. and pl. Shall is much less commonly used, but is often used in first-person offers and occasionally for emphasis.
I shall/will see.
You will see.
He will(shall) see.
We shall/will see.
You will see.
They will see.
As the past of will, for example in indirect speech.
"The next meeting will be in a month's time" becomes
''He said the next meeting would be in a month's time''.
Polite requests and offers (a 'softer' form of will)
Would you like another cup of tea?
Would you give me a ring after lunch?
I'd(I would) like the roast duck, please.
In conditionals, to indicate 'distance from reality': imagined, unreal, impossible situations
If I ruled the world, every day would be the first day of Spring.
It would have been better if you'd word processed your assignment.
After 'wish', to show regret or irritation over someone (or something's) refusal or insistence on doing something (present or future)
I wish you wouldn't keep interrupting me.
I wish it would snow.
(This is a complicated area! Check in a good grammar book for full details!)
Talking about past habits (similiar meaning to used to)
When I was small, we would always visit relatives on Christmas Day.
Future in the past
The assassination would become one of the key events of the century.
Can & Could usage:
Talking about ability
Can you speak Mandarin? (present)
She could play the piano when she was five. (past)
Can you give me a ring at about 10?
Could you speak up a bit please? (slightly more formal, polite or 'softer')
Can I ask you a question?
Could I ask you a personal question? (more formal, polite or indirect)
Could is used as the past of can.
He asked me if I could pick him up after work.
You can drive when you're 17. (present)
Women couldn't vote until just after the First World War.
Choice and opportunities
If you want some help with your writing, you can come to classes, or you can get some 1:1 help.
We could go to Stratford tomorrow, but the forecast's not brilliant. (less definite)
Could (NOT can) is sometimes used in the same way as might or may, often indicating something less definite.
When I leave university I might travel around a bit, I might do an MA or I suppose I could even get a job.
I think you could be right you know. (NOT can)
That can't be the right answer, it just doesn't make sense.
If I'd known the lecture had been cancelled, I could have stayed in bed longer
I think you should go for the Alfa rather than the Audi.
You shouldn't be drinking if you're on antibiotics.
You shouldn't have ordered that chocolate dessert - you're not going to finish it.
Obligation: weak form of must
The university should provide more sports facilities.
The equipment should be inspected regularly.
The letter should get to you tomorrow - I posted it first class.
Things which didn't or may/may not have happened
I should have renewed my TV licence last month, but I forgot.
You shouldn't have spent so much time on that first question.
Ought to usually has the same meaning as should, particularly in affirmative statements in the present:
You should/ought to get your hair cut.
Should is much more common (and easier to say!), so if you're not sure, use should.