Hi Hillary!
The Dining Plan covers you for each night of your stay. So if the 9th is the first night of your stay, your Dining Plan is activated on that 1st night. But the good news is: you don't lose those credits even if those credits aren't used the 1st night. Disney tries to split it up, so families are covered (translation: fed) for their entire stay. But you can use as many of your credits at any time until they are all used up. Since it appears you'll have extra credits from that first day, you can look to perhaps reserve an extra dinner show, or somewhere that takes 2 table service meals, like Cinderella's Royal Table. If you have table or quick service credits left at the end of your trip, some quick service restaurants will allow you to convert that 1 credit into 3 snacks (just need to note this is at each restaurant's discretion to do this).
For quick service, it depends on which plan you selected. If it was the Standard Dining Plan, that comes with 1 quick service meal per person per night. So 1 person staying 9 nights should have 9 quick service meals.
On the Deluxe Dining Plan, you are entitled to 3 meals per person per night, and they can be either table or quick service. So technically, if 1 person chose quick service every meal on a 9 night stay, that would be 27 quick service meals!
Have an extra fantastic time on your upcoming trip. The longest I've stayed at Disney at one time is about a week, so 9 or 10 days sounds extra phenomenal!!