Hello Keron

For plain future tense (that is, mere statement of intention), the first person (1 or we) takes shall.  For example, 'I shall go home' if 'I' was imply affirming my intention of going home in the future. 

For plain future tense, the second person (you) and the third person (he, she, they, or a pronoun) takes will.  For example, 'You will go home.'

'I will' or 'we will', on the other hand, is not used for a plain statement of intention, but only when there is a choice or volition.  'I will go home' meaning that if I had the choice of going or not going, that I really wanted to go home, that I had considered the possibility of not going home but discarded the idea.  However, today, it is accepted that when using the first person will and shall may be interchanged.

When using other persons (you, he, they) will is still usually used unless a threat or a promise is implied:

'You shall pay for this!'

'They shall get the reward!'

The words 'should' and 'would' are the conditional forms of 'shall' and 'will'  and the same rules apply.

If I said, 'I would (like to) go home.' it means that, given the choice of staying or going home, I should prefer to go home.

'Should' can also mean 'ought to'.

To my understanding, will and shall are used in future tense of course.

But would and should aren't only used in past tense.

Sometimes "would" is a polite way to give an ask question:

Would you like to have your dinner with me?

"Would" be applied to Subjunctive If Cluases:

If I were rich, I would buy a Benz.

"Should" when not using to talk about sth. past, it means "ought to"

You should finish your assignment after class.

No one should commit crime.

"Would" and "should" have the meaning of "past" when their usage like this:

I would have done the essay.

I could have done the essay.