Spice,
Ther are several things that could cause harshness in the ride. Most have already been mentioned in this thread.
Easiest to most difficult to check:
*Extremely worn stock tires (the ride on my worn stock Kumhos was unbearable)
*Tire pressure should be not much more than recommended, especially on work Kumhos.
*Shocks and struts (just becasue you are riding at stock height, does not mean that the previous owner had it at tock height. This could have casued the shocks/struts to wear out prematurely) If you experience a single harsh "hit" with nor shimmying afterwards, it could just be attributted to the tire/wheel combo. If you get some shimmying or receding vibration it could mean worn shocks. Worn shocks will over-compress and basically hit "bottom" much more easily. Our shocks/struts are already weak from the factory so it doesn't take much to fully compress them, especially with our weak-from the factory springs...
*Bumpstops - If the shocks were used with lowering springs, there is a very slight chance the bumptsops were cut. (Some spring manufacturers recommend cutting bumpstops, and although I have not heard of this with Velosters, it might be worth a check.
Also might be worth noting where the harshest sound/most violent action takes place when you hit that bump. Is it in the front or the rear?
To this day, if I hit a pothole or road indentation at the wrong angle, it feels and sounds like the struts are going to go through my hood.
Things that will fix this:
*Smaller wheels with a higher sidewall tire. 17's with 225/45/17 are a popular choice. Our stock wheels are heavy (as previously mentioned) so going to a lighter wheel will also help. Careful not to go to too large a tire as your speedo will be way off and your tires' sidewall might rub on the fenders. Don't ask how I know, please...:disappointed:
*Larger tires with stock wheels: I am currently riding on 18X8 Klasse wheels with 225/40/18 and what a difference this size tire makes! I still cringe everytime I go over a "Bad Bump" but generally, the tires soak it up well and I cringe for nothing. Perhaps when the tire wears more it will be a different story.
*Upgraded suspension: There are several ways of doing this but the right way is to replace both shocks/struts AND springs. In fact, after using both lowering springs and decent quality coilovers, I would go coilover (as long as its affordable) The great thing about a quality coilover is that the strut/shock damping and the spring rate are already matched for you so there is no guesswork. You just bolt it on, make sure your are at the rideheight of your shoosing, get aligned and voila! You are done!
Another good way of doing this and also recommended is just getting a good set of adjustable shocks (KONI comes to mind for the Veloster) These are also a great purchase if you don't have the $$ for a set of decent coilovers. This allows you to run stock ride height and match the damping of the shocks/struts to the OEM springs (better than Hyundai did). Also, if later on, you have the money and want to ride lower, you can get a set of lowering springs to match the height you want and ajdust the damping to the new lowering springs. This combo tends to ride smoother than cheaper coilovers.
On a side note, there are some things you can do to fight the bumpsteer. Rigid collars for the under-body support, and Pierce motorsports has a couple of products that are highly beneficial to reducing bumpsteer, their torsion bar was mentioned above and I have one. Bracing for the body, both at the strut tower and beneath the engine are also helpful in stiffening the chassis. However, remember that the more stiff the chassis is, the more uncomfortable it will be.
In short, the biggest changes will come from improved suspension and tire choice.