Learn Finnish: Sinun pitää muistaa kaikki tulot ja menot. - You have to remember all the income and expenses.

This is the core rule for obligation sentences! When using pitää to mean "have to" or "must", the person who has to do the action is put in the genitive case (sinä becomes sinun). The verb pitää freezes in the 3rd person singular form and never changes to match the subject. The actual action you are doing (muistaa) stays in its basic dictionary form.

Since tulot ja menot are the objects of the sentence, why aren't they in the partitive or genitive case?

Because this is an obligation sentence! A major rule in Finnish is that the object of a necessity verb like pitää or täytyy is never put in the genitive case. Since "all the income and expenses" is a complete amount (a total object) and is plural, it takes the nominative plural form, which simply ends in -t.

Why doesn't the word kaikki have a plural -t ending to match tulot and menot?

The word kaikki (all/everything) is slightly irregular. In the basic nominative case, its plural form is exactly the same as its singular form: kaikki. It does not take a -t ending, even when it modifies plural nouns like it does here.

I recognize the roots of tulot and menot. Are they related to the verbs for "come" and "go"?