Hi Corey,
And welcome!
Well, I'd have to ask a question or two, before giving any suggestions.
First, what type of track are you going to be racing on?
If we are talking about autocross and fairly unmodified cars, that's one thing. If you want to go road racing, that's another. Drag racing is yet another.
My TR4 used to compete in the same class as Corvettes, in autocross. Obviously the big, V8s would blow my doors off on a road race track.
For a simple autocross setup, I'd suggest an older car, like the TR4, with it's ladder chassis and live or solid rear axle. Much easier to set up, although the ride won't be as nice and cushy on your daily commute. The engine can be tuned for autocross with original SU carbs, quite satisfactorily. Webers and such don't give much advantage in this type racing.
On a road course, the Webers can really do their thing and give an extra 5 or 10% at the top of the rpm range. But, figure on swapping in a hotter cam, doing porting, increasing valve size, installing a header, maybe changing to an electronic multi-spark ignition, etc., etc. to get the carbs to work at the top of their capabilities. It's really a system to get the motor to breathe as best possible at higher rpms, and the more it's improved at that end the less enjoyable it will be as a daily driver.
When it comes to exhaust, you are faced with similar questions. For road race, a large, 2-1/2" exhaust system with minimal muffler and restriction will allow the maximum top end power. But, for autocross a 2" might be a better choice, to retain off-the-line acceleration and good torque at in a lower rpm range.
Tuning and tweaking can be done a little at a time.
As already noted, TR6 are harder to set up in terms of handling, due to the more complex independent rear suspension. But it's possible to improve them a lot, if you have the time and the money!
I'd suggest you try to get a copy of Roger William's book, "Improving TR5, 250 and 6". There are tons of ideas on setting these cars up and the information is really current. He's also written companion books about restoring TR6 and several pertaining to both restoration and improvement of later and earlier models.
Perhaps the easiest way to boost horsepower in a TR6, to the levels you want, would be to install fuel injection. The British models of these cars with PI (petrol injection) were around 150 HP right from the factory. Williams talks about various options to make the conversion in his "Improving" books.
The "over the top" way to make big horsepower in these little old cars is to drop in a V8. Again, Williams shows a couple examples and might give you some ideas.
I'm starting to sound like an ad for Roger's books, but I really have no connection!
You might also want to be a little selective about the age of car you choose. I see you are in California like me and if you want to drive the car on the street, there are more limitations here than anywhere else. My car is a 1962, pre-pollution control, so I don't know much about this. I suggest you do a little research in that respect, too.
You might also want to see if there is a Triumph club in your area, and go as a guest or join, to see what folks have done with their cars, maybe get some up close and personal advice and inspiration.
In terms of racing, the first thing to do is pickup a rule book for the type of racing you want to do, to get guidelines of what modifications are allowed in various classes and what safety equipment is required. SCCA has a website with rule books for their various forms of racing, which probably act as a guide for many racing bodies in the U.S. They call autocross "Solo" and have it set up at several levels of preparation, to allow street cars to compete in one, while more modified cars go head to head in another.
Hope this helps!
Alan