If you want to do this on the cheap, goto Ebay and search USDRRT 17 and you might find some take offs of DOT rated race tires. This tire dealer is extremely good and stands behind these used tires. I have used him for years and usually purchase used spec miata tires for really cheap for my Escort and now my Miata. I used them for autocross and track days. I even bought a set of used formula tires for 13" wheels i had and found the limits of grip to be scary! Currently there are a set of BFG's for less than $200 plus shipping. This way you don't spend alot, you should get a couple of track days off them and then you can make big decisions on where to go from there.
Now, with great powers comes greater responsibility! (thanks stan lee). Grip is everything good and everything bad.
First the bad....There are several folks out there that run Cobalts that can clue you in to the possible weak points of the vehicle. Mustangs with 335/17 tires will break caliper brackets, Neons with 275-35r15 snap axles, and I know someone that broke a rear end from the added grip on a Corvette Z06!!! So if this is your daily driver that you are going to try to drive home after a track day, make sure you have a backup plan. Also, race tires tend to be edgy at the limit. In other words, they don't make as much sound as regular street tires so exceeding their limit can happen without warning and put you into contact with immovable objects. SO work up to the potential of the grippy tires carefully. And braking will be better, so you will be able to brake later than most people on street tires, so be careful as they stop sooner while you try to stop later and bang goes your front end to their rear!
Now the good... You will corner like a real race car(sort of), and start to carry your speed through the corners and catch even more folks down the straights. You will be able to brake harder and deeper making you feel like you started at the back of the group just to prove you can pass the entire field. Assuming the car has a LSD, plant your right foot to get the car to pull through small errors. The art of driving a FWD is just that an art, and with better tires you will start to enjoy the experience even more. I always had fun with the Escort passing Camaros, CTS-V, Mustangs, Porsches....they never knew a econo box could be fast, or they were just that new. Still I got to brag with photos to prove it (I have been club racing for over 25 years).