Could you please illustrate when I would use nowy and when I would use nowego?
The endings of polish nouns, pronouns and adjectives change depending on the part they play in the sentence. The form of the word you find in the dictionary is called the nominative case and is used for the subject of the sentence. The direct object of the sentence uses the accusative case. To answer your question, we need to mention another case, called the genitive, whose basic function is to express composition or possession ("of" in English), but in some ways overlaps with the accusative.
So
nowy, "new" has nominative
nowy, accusative
nowy, genitive
nowego.
samochód, "car" has nominative
samochód, accusative
samochód and genitive
samochoduMam nowy samochód. "I have a new car"
nowy samochód is accusative because it is the direct object of the sentence.
Koszt nowego samochodu to 50 000 złotych "The cost of the new car is 50 000 zlotys"
nowego samochodu is genitive because we are talking about the cost
of the new car.
So far, so good. Things start to get more interesting when you want to negate the first sentence. For the direct object in a negated sentence, the genitive takes the place of the accusative
Nie mam nowego samochodu. "I don't have a new car."
Another way in which the genitive overlaps with the accusative is as follows. As we have just seen singular masculine nouns and their associated adjectives are identical in the accusative and nominative cases
if they refer to something which isn't alive(human or otherwise), desirable or interesting, when the accusative is identical to the genitive instead. Let's deal with "alive" (animate) first to show the general idea.
Kot "cat"
Mam nowego kota. "I've got a new cat"
kota is still the direct object like
samochód above, but because it's alive its accusative case endings are the same as the genitive.
When I said desirable and interesting things behave in the same way, I was being slightly tongue-in-cheek, but it's true that there is a whole bunch of masculine nouns which behave as
if they refer to living objects. Examples are iPhones (mam nowego iPhone'a), mushrooms, makes of cigarettes and cars, waffles ...