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I'm doing the Pimsleur Russian audio course and I've come across some sentences that I'm not sure about.

Example 1:

I have money = у меня́ есть де́ньги.

So far, so good. "At my place, there is money." It seems intuitive that money is in the nominative here. But then:

I have enough money = у меня́ доста́точно де́нег.

Although the sentence is almost the same the money is now in the genitive.

Is that really the case here or am I misidentifying cases?

And then Example 2:

Can you give me a little money? = Вы мо́жете дать мне немно́го
де́нег?

This is really confusing, I'm not sure I'm hearing it right. But it sounds like here money is also in the genitive. But I expect it to be in the nominative because "give something" and the something would be in nominative. Or what am I missing?

Большо́е Спаси́бо for your help.

You got the cases right in all three sentences.

I'll try to provide English translations which would be as close to the literal meaning of the Russian phrases as possible.

Please note that they are not actual translations, they are English approximations of grammatical structure of the original Russian phrases.

У меня́ есть де́ньги. // There are money next to me.

Russian uses proximal possession: Russians don't say "I have something", they say "there is something next to me" to convey the same meaning.

In English, you would have said "I have him" but "he's next to me".

If you look closely you can see that you use "him" in the first sentence but "he" in the second one.

This is because grammatically, "him" is an object in the first case, but a subject in the second case. Similar logic applies to Russian, except that in Russian almost all nouns decline, not just personal pronouns.

У меня́ доста́точно де́нег. // I have enough of money.

In English you would normally use "I have enough money", but there are cases when you are actually using "of": "I've heard enough of this nonsense".

If we go back to the personal pronouns analogy, you would have said "I have enough of him" in English. "Of something" mostly (not always, but mostly) translates to Russian genitive, and that's why you're seeing it here.

Вы мо́жете дать мне немно́го де́нег? // Can you give me a little of money?

Same logic as above here.

But I expect it to be in the nominative because "give something" and the something would be in nominative

In Russian, a direct object to дать ("to give"), or to any other verb for that matter, is never nominative. However, it's not the reason денег is put into genitive here. The real reason is that it's not денег that is the direct object, but немного.

Денег complements the немного and it's the latter which governs the case of the former. And немного requires a genitive.

I expect it to be in the nominative because "give something" and the something would be in nominative

This is wrong, "give something" goes with accusative, not nominative. It's actually not wrong to say "дать деньги" for "give the money", when you refer to a predefined sum of money identified by the context.

Additionally, there are quantifiers used with mass nouns, such as "достaточно" or "немнoго". These require genitive (as you have seen), independently of the verb you used:

у меня́ есть де́ньги -> у меня́ доста́точно де́нег (nominative -> genitive)

Вы мо́жете дать мне де́ньги? -> Вы мо́жете дать мне немно́го де́нег? (accusative -> genitive)

де́ньги may be a poor example because it has the same form in nominative and accusative.

You undestand it right.
You just need to ask the right question to understand.

It's easy here:

"I have (what?) money" - "У меня есть (что?) деньги"

And a little bit more complicated here:

"I have enought (of what?) money" - "У меня достаточно (чего?) денег"

Same thing with "little". Because when you give an additional word to explain "how much", it's always "how much of what", not "how much what"

To complement Quassnoi’s answer:

У меня есть (что?) деньги^NOM.

У меня [есть] (что?) достаточно[e количество^NOM] (чего?) денег^GEN.

Вы можете дать мне (кого? что?) кучу^ACC(чего?) денег^GEN?

If you need a system knowledge

First, learning cases - is nearly the most complex thing you will find in Russian (others will be simplier!), because the cases mechanism is the thing that sets the subject of every sentence (subject is always nominative) and allow to place other nouns in much more flexible way than in English.

This is a full table of cases from every Russian student book, with single-word examples in single and plural forms. To understand all these questions, you need to search for examples about cases in Russian - it is more simple than lucky you must be to find similarly fulfilling information in English. This is a kind of such boring topics, that are learned by long practice.

The following can be where to start from, although it is not really full.

Good luck and don't give up!

Now, I'll explain exactly your example, with more examples. I don't know all cases in English, so I'll name them in Russian. The single form will not be provided, because 'деньги' has only plural form in Russian.

Именительный падеж

У меня есть (что?) де́ньги = I have money

Родительный падеж

Мне не хватает (чего?) де́нег = There is a lack of money for me

Дательный падеж

Радоваться (чему?) деньга́м = To be glad with money

Винительный падеж

Я нашёл (что?) де́ньги = I have found money

Творительный падеж

Он заплатит (чем?) деньга́ми = He will pay with money

Предложный падеж

Главное, не думай (о чём?) о деньга́х! = Don't think about money!

(Note that the last case uses all possible preposition, not just 'о'):

Предложный падеж

Да Скрудж Макдак просто купается (в чём?) в деньга́х! = Scrooge McDuck simply swims in money!