Depends on your budget. Don't even look twice at the $150 "coilover" kits you see on ebay; they'll make your car bounce off the street (horrible rates, bad construction = spring sag + jammed perches). Go with a quality spring/shock combo, or if you want adjustability, go with a coilover sleeve + shock combo (ala Ground Control koni edition) or a full coilover suspension (ala Zeal B6) if you have the cash. If you do a search for suspension I'm sure you'll find what you need. There should be enough threads to cover all possible aftermarket suspension modifications .

Autocross is a good way to learn how to drive your car; I would recommend doing an autocross school (try an Evolution school) and then go to a few autocross events and see how you like it. I personally found autocross to be much more boring than, say, a high performance driving school/track day , but in the short run it's more affordable (pay 20 bucks for 3 minutes of autocross seat time versus shelling out 200 bucks for 2 hrs of road course seat time) and you'll learn more about car control overall.

How low? As low as you want . If you're lowered past 2 inches a camber kit will help even out the tire wear (although frequent tire rotation/balance should be fine). Note that as you go lower you'll want shorter springs/shocks to retain optimal handling, so getting off-the-shelf Ground Controls and slamming your car 4 inches will likely result in worse handling than a modest 1.5" drop.

You'll notice suspension mods more than any other engine bolt-on (visually your car will look better as well), so I would recommend doing that first and then moving on from there.

Shingo