Hyundai Repair: Suspension Gone, hyundai accent, cheap tires

Question
The car was doing fine until i had the front tires replaced and from driving it off the lot i noticed that when i went around a right curve i was trying to fish tail.  This progressed very quickly to having no bounce in the front when you pushed down on the front end and in the back it bounces too much.  I had the tire people (who had replaced the tires) look at it and they said the front struts and mounts were gone.  I had another car repair place look at it and they said the front and back are gone.  Now, when i drive it i am all over the road and the smallest bump will send me bouncing up to the roof.  It's very frustrating.  I have had prices of $500 for the front -- another said I needed to do the back not the front -- and they also said it would cost $1500 to have it done.  
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Followup To

Question -
My 2000 Hyundai Accent with 62000 miles on it has completely lost the suspension in the front and the back.  What could have caused this?  I've never had a car even need shocks replaced much less the struts and the mounts.

The car has not been abused -- mostly highway driving.

Answer -
Usually, this is caused by wear, but it's not typical for a car to need all four struts and mounts at this age and mileage.  How was it determined that these things were needed?

Answer
I've seen the fishtailing you describe occur with either low tire pressure or *very* cheap tires.  Before you do anything, check the tire pressures in all tires to verify they're all around 30 PSI.  If any of them are significantly lower than that, set you tire pressures and recheck to see if the problem is still there.

It's normal to have very little bounce in the front end when pushing down on the car.  Usually, you push down on the car and it'll come back up, move just a little more, and stop.  The quicker it stops, the better condition your struts are in.  The back tends to be significantly less damped, so it may bounce up and down a time or two before stopping.  The industry standard is that three bounces is okay and more than three bounces is excessively worn.  In my personal opinion, the system is usually well worn at three bounces-- enough to notice a difference, but not enough to cause a hazard or a continual bouncing motion when driving.

I find it very suspicious that everything started immediately after you purchased tires.  While I can cast doubt on the diagnosis, I have the handicap that I don't actually have your car present to inspect.  I will say that I don't follow why the second shop said you needed new front struts.  Since you say the front end didn't bounce, what was their reasoning for replacing them?  I also don't follow why either shop said you needed new mounts.

I suppose the bottom line here is trust.  There are some places that push their mechanics to sell struts and similar components to the point of replacing parts that are perfectly good.  But, like I said, I'm in no real position to say either diagnosis is incorrect.  But if you suspect either of these places may be trying to sell you work you don't need, you should find a trustworthy place to have your car checked.  If you don't have one of your own, check with your friends by word of mouth.