First, glad to see that you are doing well and that you weren't harmed. That's a tricky situation that could have easily gotten far more worse than it did.
As a few have already mentioned, you need to have your car inspected by the dealer ASAP. I would park the car tonight, call a towing company, your insurance company, whomever you can get towing through (roadside assistance) and have it towed to the dealer.
Why towed? Because with an engine fire, you have no idea what may be damaged. This could have exposed wires, damaged lines, cracked plastic, etc.... This is the best way to prevent any possibility of damage or injury to yourself. Overboard? Maybe...but then again watching the car catch on fire while you drive it to the dealer would be on you...and hard as hell to fault the oil place at that point...
I would then submit any bills to the oil lube place for payment. If they want to play games or refuse, the fact they almost burned your car to the ground would screw them in any court of law....
As for the comments about doing your own oil vs. quickie oil joints, it's all a matter of opinion. I've seen morons screw up their own oil change just as badly as a oil changing place. The dealers are actually getting much better with their pricing, and often will give a membership type freebie after so many, depending on the dealer.
Ultimately, do what you think is best. I still wouldn't drive the car until it's inspected for safety purposes.