What do you do if you accidentally threw away your birth control pills a week early and is it ok to start your new pack this Sunday as you normally would or do you need to wait until your next cycle?

Start your new pack immediately. You can start the next pack the minute you realize the rest of your pill pack has been skipped. You probably won't get a period until the inert pills at the end of the next pack--that's normal. If you skip an entire month of pills, you will be left unprotected. If you start the next pack a week late, like your questions implies, use a backup method for a week.