First off, this thread is mainly written for people with street cars who track them seldom. Those people who frequent probably know the basics by now. For those of you who want a more "sports car" feel on the way to work, and want your car to be stiffer and more rigid, below are some of the mods you can do.
Shock Tower Brace
Easiest for most cars is commonly referred to as a "Shock Tower Brace".
This connects the two shock towers together. (front left and front right OR rear left and rear right)
These can be had fairly cheap, in the 100-300 range depending on the car and greatly decrease what is known as "body roll" in turns. What the brace does is tie the car together more (just as a roll cage would) and doesnt let the car flex as much when load is put on the frame (lateral g's).
Install is straight forward on most cars for the front shocks. You should just have to open your hood and remove a few bolts, put the brace in, and reinstall the bolts. Very simple.
Here is a pic of shock tower on a Trans Am.
The next easiest and cheap modification would be larger/firmer sway bars. The sway bar is what connects the A-arms (or sometimes a solid rearend) to the chassis. The stock ones are always small and weak, as Arnold would say. A stiffer sway bar will also help in the turns by tying the car together more. The car will be less smooth over bumps with this mod because it wont allow the sway bar to flex as much because it is now larger and more rigid.
Sway bars can be had (depending on the car) for 200-1000 a set. Most cars will fall in the 300-400 range for both front and rear sway bars.
They are often offered with either polyurethane or rubber ends. Simply put, rubber will offer a smoother ride and be quieter. Polyurethane will offer a more sporty ride, and may squeak. They need to be lubed every so often, usually once a year and is very easy to do.
Installation varies vastly on the car.
Here is a pic of a set of sway bars both front and rear
Shocks/Springs (or coilovers)
Shocks and springs are always thrown around when wanting to make a car handle better. Simply put, stiffer shocks/springs will not let the car weight trasnfer so much in the turns so the car will stay more level. This is critical because you want as much contact with the road from your tires, and if one side is planted and the other side is lifting, this is making the side that is lifting "lighter" so you cannot accelerate out of the turns as well as you could if you had the car nice and level. A nice combo of shocks and springs will help do this.
Springs will help to LOWER the car. This is key as you want to get your center of gravity as low to the ground as possible while still allowing your car to go over speed bumps and get in/out of driveways.
Why not get coil overs? There are many different combos available for all cars however some have the choice of Coil overs for their car. A coil over is called this because the spring looks like a "coil" and it goes "over" the shock itself. So the shock lives inside the spring. This is exactly what you want because having the shock in a different location than the spring is never ideal when handling is in mind, however its just fine on an suv or sedan where its not as critical.
If your car currently doesnt have coil overs it still might be able to have them. Coil overs are typically more expensive than a regular shock/spring combo if thats what your car uses. However the coil overs sometimes allow for very easy ride height adjustment with a few simple tools.
Here is a pic of LG Coil Overs
Wheels and Tires
One if the easiest (and usually most expensive) ways to make your car handle better is a wheel and tire combo. 99.9% of cars come with either an all season and low speed rated tire OR some sort of runflat tire and both of these tires are meant for 50k miles and give nice ride comfort do to the tall sidewall on them.
The best thing to do is to increase your contact area as well as decrease the size of the side wall. Obviously, wider = better. However you cant just put a rubber band thick tire on your stock 15" or 16" wheel, that would look ridiculous! So most people step up to at least a 17", commonly 18" or 19" wheels. This is where it can get expensive.
Tires tires tires tires. This cannot be stressed enough. All season radials are great...not for handling. Most true enthusiasts will have a summer set of wheels/tires, and then a crappy set of rims with snow tires for the off season if they live in areas that it snows. Ideally you want a nice low profile tire, that is nice and wide (how wide depends on the car) and something that is a bit sticky and will have a lower tread life due to this softer compound.
A picture of large wheels with low profile tires on a Corvette