After an accident, what next?
Report it to his insurance company first. Period.

If for whatever reason the driver decides to renege on his admittance of fault and declines to pay damages to your bike, you're basically screwed if you're past the insurance company's reporting threshold. However, do not sign any paperwork the insurance company asks you to (which may limit their liability or waive some of your rights) and inform them that you're having a bicycle shop prepare a quote for damages.

After you've contacted the driver's insurance company, take your bicycle to the nearest shop and have them fully inspect the bike and quote you for any necessary repairs. Ask them to be thorough and include the cost of absolutely everything that may have been damaged in the crash: wheel re-truing, handlebar tape rewrapping, etc.

When the bike shop gives you a quote, give it to the person who hit you. If he reimburses you, then you're done. If not, you'll have to work through their insurance company. It's probably best at that point to contact your own insurance company and see what steps they will take to help you receive payment.