Tires are the most important aspect of autox. I always think of this cliche. A chain is only as strong as its weakest link. So you can have every mod under the sun, but if you have crap for tires, you have no way to put the power to the ground.
If you have the dough and an extra set of rims to devote to the track, get some R compound tires. Hoosier S05's or Khumo v710's. They are allowed in stock class and will make a world of difference.
Your next question was about the suspension. Where you go with that is all gonna depend on what class you want to be in. If you wanna stay in stock class, you're only allowed to upgrade the struts and front sway bar. No sprins or coil-overs allowed. And of course, you can adjust your alignment.
You aren't allowed to do much in stock class, but that's not always a bad thing. You can greatly improve your cornering stability and reduce body roll with a decent set of adjustable struts. You can also reduce some understeer with a little negative camber to your front wheels.
I'd recommend you try to find some autocrossers who use a similar car to yours, and ask them their opinions on their set-up. You'll probably gain more knowledge that way.
But, if you need a place to start, new sticky tires are a must, and some adjustable struts (koni, kyb agx, tokico illumina) are always the way to go.
Best of luck!