When you connect your iPhone to iTunes on your computer and go to the summary page for the iPhone, one of the buttons visible is a backup button. Tap that button and iTunes will create and store a full backup of your iPhone. Then, when you connect your new iPhone to iTunes, you'll be given the choice of setting it up as a new iPhone or from a previous backup. Choose the backup of your old iPhone and everything in the backup will be installed on your new iPhone. This should include all installed apps along with the app data, your photos, music, movies, and TV shows as well as your contacts.
You don't have to repurchase any of your previously purchased apps, they'll be installed as part of the restore from your backup. Any previously purchased apps not currently on your iPhone are available for free download from your App Store purchase history. This also applies to all previously purchased iTunes content, which is available in your iTunes purchase history.