I could go to the beach with you but I should stay home and study ( I might want to go to the beach but I have to study) I would go to the beach with you but i should stay home and study (I want to go to the beach but I have to study)

could means an ability to do something for example: Could you give me a hand? Could I stay here? I could ride a bike etc so here one has the ability to offer help and do something, it has both possibilities of acceptance and rejection Would: shows one's will to do something, more formal, polite expression for example: Would you like a drink? -- shows offer, if one would want something to drink Would you help me/open the door for me? -- ask for help, shows one's will of asking someone to help I would like to have a book please -- I want to have something, but more formal and polite Should: showing one must/have an obligation to do something, quite demanding, commanding word, depends on what context for example: You should revise now. -- You have to revise in order to achieve high grades./ One must do something We should help save the environment by reducing pollution such as using less fuel. -- Shows obligation that one has to do something in order to improve You should call your parents -- You should/have to call them as they are worried about you