1. Catherine had never flown in an airplane before.
Because "airplane" begins with a vowel sound, you need "an" rather than "a" before it.
2. Carla wanted to go to Six Flags Amusement Park with Eric.
You do not need "the" before Six Flags Amusement Park because it is a singular proper noun.
3. I want to give you an idea of what will be expected of you next semester.
4. Julie had been dancing for t he McKendree College Dance Team for almost three years.
5. The full moon will occur in three days from now.
"The" is needed before the noun here since it is referring to a specific full moon.
6. Are you going to go to Carlyle Lake for the day?
"The" is unnecessary since Carlyle Lake is a singular proper noun.
7. We are looking for an apartment that will allow us to have three dogs and four cats.
8. Peter took a piece of bread and spread peanut butter all over it.
"A" is necessary here since "a piece" makes the bread a countable noun.
9. Yolanda offered me milk to go with my cookies.
The "a" is unnecessary since milk in an uncountable noun.
10. Confidence is what Juan was striving for.
The "a" is unnecessary since confidence is an abstract noun (uncountable).
11. He wanted to rent equipment from Ryder to haul his broken car away.
"An" is unnecessary here since "equipment" is an uncountable noun.
12. Students in t he United States do not often begin thinking about graduate school until they are juniors or seniors.