I want to learn more about cases, understanding cases, when to use them and how to use them correctly.

I've been taught this (below), but I would love if somebody could tell me which case is being used and why. Or better yet, direct me to somewhere on this site or just recommed a site or web page to look at that would help me. Just a little help or a lot of help is great, I'll work with whatever amount of help I can get.
(I'll use Pencil and Car for my example though)

Я не уверен...

карандаш, машинА
карандаш, машинУ
карандашА, машинЫ
карандашУ, машинЕ
карандашЕ, машинЕ
карандашОМ, машинОЙ


BUT, I remember my friend Алина helping me with Cat and Dog, which was in a different way then above. AND she wrote out the cases in Russian for me but I dont know exactly how they translate.
This is what I ended up with:

ИП - собaкА, кошкА
РП - собaкИ, кошкИ
ДП - cобaкЕ, кошкЕ
ВП - собaкУ, кошкУ
ТП - собaкОЙ, кошкОЙ
ПП - собaкЕ, КошкЕ

I suppose "подеж" means "case"?
I know Im pretty mixed up with this, so thank you very much for your help!


So!..
I know that "i have a car" translates to "Я имею машинУ."
And I think "My mom (mama) is at the store." is "Моя Мама в магазинЕ."

I still feel that моя Русскaя грамматикa is a little messy and I would love to be able to get more organized with my learning, especially my grammer.

Спасибо!