He's keen! 
any body can write that sketch
OP, It can certainly be done. But it's not quite as simple as one first imagines it, especially if one shows little or no experience in many of the sciences required for this very complex application. I've got over 10,000 lines of code in my arduino automatic transmission controller (4L80E) and have used almost every port in my Due. If you think about, an automatic transmission, is similar to a manual (except when at a standstill). The clutches/bands in the automatics are controlled by hydraulic servos which are in turn controlled by electrical solenoids. I race both automatic and stick cars so I have background in the area.
So, I'm going to treat this as a hypothetic case cause I have some doubts that OP will succeed in building this.
You would need a very fine servo control for the clutch activation.
For racing, the scariest part of this application is downshifting. There needs to be many rules built into the application to insure that downshifting does not upset the vehicle. You will need to couple many feedback sensors, including engine RPM, wheel speed, g-force (yes, you need g-force. if the tires are already at their limit of g-force, any extra stress from shifting will cause the car to lose control) throttle position, engine load (boost/vacuum). Clutch activation rates will need to be very closely coupled to all the sensors I mentioned above. Safety code must be in place to not allow downshifts at certain engine RPMs to avoid money shifts.
You will need to closely couple throttle override while applying the clutch to avoid shocking the system or stalling.
batuning:
for racing
I've raced on dirt tracks when young and foolish.
It would be wise to check w/ your local track to ensure your modification will be approved in any division you are planning on running.
What is going to happen when the Arduino suddenly reboots? The dude drafting you is going to be picking metal gear teeth out of his face (if he lives after he plows up your tail.)
As a general rule forum members are shy in discussing Arduino use where human life could be lost. I understand you are not concerned about your wellbeing, but others on the track may be more concerned if they found out you cannot program this yourself.
But, it is my belief that an Arduino Due could probably pull off the stunt... maybe even a Mega2560. But your starting point needs to be a great deal of research... how do those Arduino flight controllers work... what makes 'em tick?
Here is the answer and the high-level answer to your problem:
PID
We are fortunate the "help" on this board had no input to Wilbur and Orville Wright. If those pioneers listened to the naysayers in their time we would still be tied to ground travel. Best wishes in your quest. I am a newbee to Arduino so everything is new to me. Also going for a paddle shift of a manual trans. When I solve the many issues I will post. From what I have read here, this board is not the place to look for help.
lanczagato:
We are fortunate the "help" on this board had no input to Wilbur and Orville Wright. If those pioneers listened to the naysayers in their time we would still be tied to ground travel. Best wishes in your quest. I am a newbee to Arduino so everything is new to me. Also going for a paddle shift of a manual trans. When I solve the many issues I will post. From what I have read here, this board is not the place to look for help.Well, I guess you need to find another forum that will help you accomplish your plan.
One that would be very involved for an experienced coder, let alone a newby.
the major car manufacturers expend thousands of man-hours of highly qualified engineers trying to do things like this.. it isn't easy.
how much spare time have you got?
regards
Allan
Hi,
One thing I can't find rereading this thread is, what brand/model/type of gearbox the OP wants to convert to sequential shift!!
Tom.... 
Could you buy such a clutch and gearbox ( eg Renault make them ) and bolt that into your project?
A big mechanical job, but may be easier in the end...
regards
Allan