Third-party cover is insurance that covers your legal liability, i.e. your liability for any damage you cause to others with your car. Please note that cover is provided only if you are actually legally liable. If you own a car, you are required by law to at least take out third-party liability insurance for it. Third-party liability cover does not cover damage to your car. If you want cover for damage to your car as well, you should take out Third-Party Liability + Limited Cover or Third-Party Liability + Comprehensive Cover.
Submitted everything? Then take out the car insurance. You choose when the insurance takes effect: immediately or later.