Here's a mnemonic rhyme that I made up a couple years ago to help people understand the basic functions of noun cases, and also to learn the case-endings for "Type 1" feminine nouns, using девушка ("girl") as a model. The words or syllables in blue refer to the names of the cases, and the rhyming words (in green) help to remind you of the feminine endings (in red).

"Nom, nom, nom," said the девушка,
As she ate up all the baklava.
[The nominative case is used for the subject of a verb/sentence. In this case, девушка (girl) is the subject of "said".]

Shy guys generally flee
From a девушки.
[One of the most common functions of the genitive case is to show movement away from someone/something.]

Commander Data gave the tray
To the девушке.
[The dative case is used for indirect objects. Here, "tray" is the direct object of the verb "gave", and "the girl" who received the tray is the indirect object.]

I threw my shoe
But my aim wasn't true
And I accidentally hit the девушку!
[Here, the accusative is used because "the girl" is the direct object of the verb "hit".]

The old cowboy
With his девушкой
Played an instrumental version of "Ode to Joy".
[The instrumental case has a number of functions, but one is to show accompaniment -- that is, hanging out or doing something with another person.]

President Clinton spent his day
Thinkin' 'bout the девушке.
[The prepositional case gets its name because it ALWAYS has a preposition before it. Here, the preposition is "about" -- in Russian, "about the girl" would be о девушке.]

Hope this helps, at least a little!