Here's the breakdown:
Speedometer Accuracy -
Generally most people who upgrade to larger wheels will get tires with appropriately shorter sidewalls such that the overall diameter will remain very much the same as the stock wheel and tire combination. This way the circumference will remain the same and will allow the speedometer to still be accurate. These calculations are usually performed by the wheel dealer/tire installer.
Better Handling -
Larger wheels means shorter tire sidewalls therefore less lateral sidewall flex and roll when cornering. The reduced lateral roll in the sidewall will allow the steering response to be faster and more precise. For example take a yardstick and grip it at the ends and flex it. Now flex it with your hands only 6 inches apart and see the reduction in flex when the same force is applied. now picture this flex occurring in your tire sidewalls every time you turn the steering wheel. The time between turning steering wheel, flexing the tire sidewalls and when the tire contact patch on the surface reacts will affect what the driver will percieve as steering response time. The faster the response (the less the flex) the better.
Reduced Weight -
This depends on the wheels that you select. Even among 17", 18" wheels there are great differences with regard to weight (some are heavier than stock and some are lighter). Less weight than stock means reduced unsprung mass. In simple terms this means that you engine will need to expend less energy to get the wheels themselves rolling. (It's like using your hands to roll a beach ball versus a boulder). Less energy expended to make your wheels roll means more to make the rest of your car move. Alternately, it can mean greater efficiency (better gas mileage). On the contrary heavier wheels (most of the chromed variety) will slow you down.
Reduced ride quality -
The drawback to the shorter sidewalls means less rubber to cushion the ride. Minute road vibrations are first transmitted through the tire contact patch on the pavement, through the tire sidewall, into the wheel, through the suspension, and then up into the chassis. Stock wheels and tires with their large sidewalls are able to absorb many of these vibrations that are oftentimes of too high a frequency/low amplitude that the suspension is not able to dampen. The shorter and by virtue stiffer tire sidewalls with aftermarket wheels as described in the handling explanation thus diminishes this vibration absorbing capacity.
Appearance -
The current "tuner" trend tends to deem cars with the largest wheels and shortest sidewalls to be the best looking. In stock form with most sedans/coupes the gap between the fender lips to the tires is approximately the same as the sidewall highet. SUVs on the other hand tend to have fender lip to tire gaps that are greater than the sidewall height. However as one increases the wheel diameter and the sidewall gets thinner the fender lip to tire gap will remain the same. This will create an "unbalanced" or "out-of-place" look. Even with 17" wheels this dicrepancy between the large fender to tire gap and short tire sidewalls can lead to the illusion that the car is actually raised up (like an SUV). Therefore many with larger wheels elect to install lowering springs to equilize this difference.
Something else that can affect the appearance and may be vitally important to the overall safety/driveability of the car is wheel offset. Lowriders with the wheels which often times stick out from under the fenders would be an example of low offset wheels. High offset wheels tuck under your fenders closer to the "midline" of the car. The Legends requires wheels of offsets greater than +42mm depending on the wheel width. Offsets less than this will cause the wheels to stick out from the sides leading to an ungainly "truckish" look. Low offsets will more importantly negatively impact the handling characteristics and suspension geometry of the car. Having greater mass further out on the suspension arms means that the suspension will have a greater burden to bear. This is the lever arm principle from high school physics.
Hope this clears things up for you.