My experience is that it should be springs (dampers) first and sway bars second as a fine tuning element. As stated above, swaybars are good as a fine tuning element but not as the primary method of improving handling. Springs allow the car to resist roll by holding the car up, and dampers aid/modify the rate that the springs compress or rebound by adding mechanical/fluid resistance via the valving. If you could tune your car for a specific track or situation you'd dial in the springs and dampers and may not need a swaybar. The limitation is that you'd need very stiff springs and aggressive damper settings to handle the cornering which would make the ride uncomfortable (not an issue for racing). So here swaybars come into play bc they only become part of the spring forces when cornering and have little effect on straight line ride comfort. Ideally the car springs and dampers are matched to whatever purpose the car is being used....but even then the maybe a desire to slightly alter (stiffen) the handling/cornering depending on traction conditions, road, weather, and driving style.
If you rely on the swaybars for a large portion of your spring rates, then you'll need a fairly large bar (stiff)...this has its limitations. The basic result of a larger rear bar is that it transfers more force and at a faster rate to the opposite side of the rear axle and to the outer front tire. This makes the car feel much more stable, aids turn-in feel, and increases the rate of weight transfer without as much roll...feels more like a go- kart. As speeds increase the bar starts to try to lift the inside rear corner, and further loads the outer front tire. The front tire will gain more downward force/grip until it exceeds the capability of the tire then it will push (understeer)....and this setup will increase cornering speeds up to the limits of the tire/slip angle. At the same time the inner rear tire starts to lift thus reducing rear traction (gradual loss of a tire contact reduces grip) and by doing so induces more and more rotation (oversteer). By increasing and adjusting the rear bar the car can be "dialed in" to suit the driver and the road/surface conditions.
So what are the draw backs? As previously mentioned the rear bar will start to lift the inner wheel and at some point the loss of grip/traction can drop enough that the outer tire can not provide enough grip for the car to remain on course...and it "snap over steers" off the road. Additionally it robs the car of suspension travel....potentially upsetting the car over bumps while cornering. It also tends to decrease suspension compliance and can affect even straight line ride quality.
Many times a simple increase in the rear bar stiffness can significantly improve the car's handling without negative effect if done conservatively. Adding a rear bar that is 1 or 2 mm thicker than stock size (15 - 30% increase in spring rate) on a stock suspension will allow the driver to find a sweet spot and reduce the roll, improve turn-in feel and aid cornering without "over powering" the springs and dampers. Unfortunately some people feel that more is better and that no roll = better handling. Unfortunately this is not the case and they put way to big of a bar on the car. While it feels great at 8/10ths or below, it starts to loose its effectiveness near the limits or in changing weather conditions, creating an overly loose rear or worse...snap oversteer.
If you are racing or autox your car then a big bar maybe ideal, or if you've upped your spring rates and dampers and added stickier tires you may need/want a bigger bar...but even then many people simply go to far.
Recommendation: If you have stock suspension and you do not race or very infrequently race, keep the rear bar increase under 3mm (~ 50% stiffer) and ideally have a bar with 2 or 3 hole adjustment. If you now are upping the spring rates and dampers as well as sticky tires then the upper limit might be to increase bar size 3-4 mm (50- 70% spring rate increase)...much beyond this is too much rear bar....remember a little goes along way.
**If you are racing these suggestions really don't apply bc the car is dedicated to a narrow band of use and weather conditions and in a controlled situation.