I don't think there are any general diminutive and augmentative rules in English. In Portuguese almost any noun can be made small or large by adding the appropriate suffix. English has no equal.
That said, there are some word ending for the diminutive form of a word. You've already indicated one, adding the letter y to the end of a word such as kitty, for little cat. There is also doggy and horsy as diminutive terms for dog and horse. However, the rule does not extend well. The word girly would not mean little girl, but someone behaving as a girl.
There is also the suffix 'ette' that is used to make some words diminutive. Examples include cigarette, kitchenette, lunchonette (small restaurant that only serves lunch). But this ending is not used very often.
I can't think of any English form to augment a word.