After an accident, what next? Cycling home in the dark a car pulled across me to enter a super market car park. Apparently the car that let him cross didn't remove their full beam so he didn't see me in the cycle lane and I hit the side of him.

Apart from the initial shock I'm basically OK. Hip &amp; shoulder feels a little bruised but thats about it from what I can tell. 

Bike for the most part from what I can see also seems mostly OK apart the handlebars.

We exchanged details etc a witness also stopped and gave me her details. And I had a friend come and pick me up as I was a little shocked. 

The driver admitted fault and seemed genuinely concerned and offered to pay for damage to the bike, it sounded like he was a cyclist himself.

I'm not planning on any injury claims as apart from a bruised/sore hip and shoulder for a few days I reckon I'll survive.

My bike is basically my workhorse for commuting and everything in between so the quickest possible turnaround with the least path of resistance to having it fixed would be the most ideal.

Should I just take my bike to a local shop have them look it over and quote me then contact the driver, I may need to pay for it to be inspected first? Or should I go via their insurance? How fast do I need to act etc. I probably won't be able to get to a bicycle store before Friday.

(I tried to add a photo of the handlebars but apparently I need 10 points first).
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.