Firmer or softer - which way do you go first?
 

Autocross - Assuming you have a balanced and setup car and are only trying to optimize things for the particular track, and have only a few chances to make a change, and only testing is one of your runs.

I.e say you want to dial out some push. Would you first decrease sway bar rate in front or increase rear?

Is the push happening on corner entry or exit?

In most cases, decrease the front bar rate. If the car is too soft overall, then you could try increasing the rear bar. You generally want to make a change on the end of the car that has the issue.

David

I had a very well set up 911 with strong LSD. If I did not get the weight to transfer to the front on a tight corner, it did not want to turn in. I had to lift, or dab the brakes. Charging into the corner never worked, and it caught me out sometimes.

Not exactly the answer to the question, I know, but it was well set up, and capable of setting a faster time than most 911s. It was easy to induce over steer, once the front turned in.

And by issue you're referring to less grip than you want, right? So in a car with oversteer you work to soften the back rather than stiffen the front.

Depends on the conditions compared to a previous known good.

If you are on a smoother or grippier surface than before, stiffen.
If you are on a rougher or lower grip surface than before, soften.

As stated above, adjust the end of the car that isn't working.

Also, my order of operations for between-run adjustment would be:
1st - peak grip - tire pressure
2nd - mid corner balance - swaybars
3rd - transition - shocks

For me when I was into this sort of thing (car would push), I decreased stiffness in front sway bar. This was for auto-x only where the speeds are not all that fast.

If your car is rather soft overall and the course is generally smooth, I would pick the option that increases the roll stiffness of the car. So I would stiffen the rear bar.

If you car is fairly stiff, or the track is bumpy, I would be more inclined to decrease the overall roll stiffness of the car. So I would soften the front.

You adjust chassis balance to match the conditions based on the overall chassis stiffness, not which end the car is having the problem. If your chassis is generally softer than optimum for the track, you want to stiffen one of the sways bars to adjust balance. If the chassis is generally stiffer than optimum for the conditions, you want to soften one of the sways bars to adjust balance.