"Now I'm not very mechanically inclined yet, so what is my best way of permanently repairing this? Should I just replace the inner tube? Should I patch it? The tires are original from 2000, so should I replace the entire tire?"
The tire is now 14 years old and if money was no issue I would replace it but it can also be easily repaired and the patch should last for years. The tube will need to be replaced regardless. Welcome to the forum.