If you are just starting out, there is no need to spend a lot of money on changing the car. At local events it is mostly all up to how good the driver is anyway.
A good set of tires is a good idea. With all the current high performance summer tires that are popular for ST you should be fine driving them at autocross and on the street in the summer so you don't necessarily need an extra set of wheels yet.
For brakes, don't try to get too agressive. You will want to get something like the Hawk HPS or Carbotech AX6 because autocross doesn't build a lot of heat in the brakes and those are better for both autocross and street.
The next big thing I would suggest is to get a good alignment. Get as much negative camber as you can in the front and somewhere around -1 degrees in the rear and get close to 0 toe front and rear (maybe a little toe-in). As you get better you will probably like toe out, but its not as stable so its not as good for daily driving.
That will get you a long way to being competitive. Spending your money on going to more events will make you faster than more parts.
Most of all, just have fun!