I too Have an 06 STI that I am starting to AutoX. Right now I am using stock rim size 17x8. Running STU classing. I am not an expert, i would say after about 10 total driving events, PDX and autoX combined, im nearing intermediate, but probably still beginner.
Anyway, based on your budget Id stay with the stock wheels for now. The stock BBS' for 06 are 17x8 and weigh sub 20 lbs each which is not bad at all. Spend your money on tires (when you feel ready).
Personally I am running 235 45 17 (225 45 is stock size) on my wheels but on the 8" rim i feel there is too much flex for autocross in the sidewall, but that is also because the tires I have are not of the product listed above (Michellin PS2). More flex in the sidewall may have some more benefit in road courses though with higher speed turns and slower transitions.
Since you are a beginner you have a lot to learn about weight transfer, tire flex, suspension dynamics, etc about the car. Not to mention getting a feel for your car behind the wheel and driving techniques in general.
I would like to pose a more specific question to this thread:
What tire size is really the recommended size for each tire product listed above on the stock wheel size (given the stiffer sidewalls over what I am running)?
EDIT: I am sure there are threads on this that I am missing. If so just say so. Sorry in advance.