You can use the following sequence to delete everything including the quotes:
da"
Keep in mind this only works on a single line, and will remove any trailing spaces after the last quote.
As pointed out by @James in the comments below, you can also use the delete inside sequence to delete the characters within the quotes without deleting the quotes:
di"
You can also use the change inside sequence to remove the characters and switch to insert mode, allowing you to easily replace text within quotes:
ci"