Yes, adjusting the suspension will have as great of affect as the other parts.
If its on a paved track then you want pretty high spring rates (about 75% of the maximum for the front and 70% for the rear). Set the shock bound to 3 or 4 for both, and the shock rebound to 10. I used to know what to set the rest of the settings, but I have since forgotten, and haven't tinkered with them enough in this game to figure them out again. Oh, and lowering the ride height helps, but sometimes going too low will make the car harder to turn.
For the braking controller, if you want stability when you brake hard (in other words no fishtailing, or the back end becoming loose) then set the front to be a lot higher than the rear. If you want to get more of a slide (kinda like drifting) then set the rear to be higher. I usually set the front to be about 24 and the rear to be about 10. Keep in mind that raising these will cause more wear on your tires, and so you might want to lower them for endurance races or things like that.
Downforce, typically the more the better, but you don't want the front or back to have a significantly higher amount than the other (as it'll cause instability). Also keep in mind, the handling ability at higher speeds will be much different than at lower speeds because the downforce will only really come into play much at the higher speeds. Also, the more downforce, the worse your top speed or acceleration at higher speeds will be. Oh, and it'll cause more wear on your tires as well.
If you load your car with horsepower its very hard to overcome its tendency to oversteer and generally not handle that well. No matter what you do, it will still handle much worse than it will with less horsepower.