It's a bit long to explain all the differences between the modals you mention. You should have a look at http://www.e-anglais.com/cours/modaux.html
In short (very short!) 1. SHALL is used in suggestions (with I/we) Shall I lay the table now? 2. WILL is used in the future simple She will be 18 next week. Shall we throw a party? 3. SHOULD is used to give a piece of advice. They should give her a present for her birthday 4. WOULD is used in the conditional. If I were you, I would do exercises on the auxiliaries and modals 5. COULD is the past of CAN and it is used to express a general ability. He could already play the piano quite well when he was five.