These are all mandatory parts. If any of them are missing you will not be able to ride, so, make sure you have them all. When you buy a new kite it usually comes with the bridal system, kite-lines, control bar, a safety leash, an air pump with a hose, and a pressure gauge. Harnesses usually come with a spreader bar of the size suitable for the size of the harness itself and an optional knife. Yet what exactly is included will vary from brand to brand, so, double-check the description and go over the list a few times to make sure you have it all.
And I did not include any luxury items like a wetsuit, booties, gloves, dry suit if needed, helmet, sunglasses, camelback to keep you hydrated, etc. You can see by now that kitesurfing in a colder climate will be more costly.