So not a Harry Potter cloak, eh?
Honestly, as cool as invisibility would be, I bet cloaks would be more bothersome than people realize. What if a tear forms? The invisibility would be compromised. What if you trip and stumble? Or if you're in a quiet area, how to control the noises of your footsteps and breathing?
Additionally, I'd be afraid of getting in legal trouble. Maybe I'd try to work as an investigator using it, to ensure I'm working with the system, not against them. Then again, teenage me might simply enjoy pulling pranks on people with the item, and keeping its existence a secret.