Cheapo tires, Chinese tires, and some "High Performance" tires from Japan will start to do this after 10,000 miles.
I had Sumitoro HTR+'s on my Alero back in the day, after 10,000 miles, they shook and vibrated like what you are saying. By 20,000 miles, they where shaking pretty bad and less than 4/32nd tread which is usually when I replace them.
A mechanic told me, "You only should buy one of these 4 brands of tire: Goodyear, Michelin, BF Goodrich, or Pirelli." My mechanic now likes Continental for all rubber products (He used a Continental timing belt set on my Volvo, which is the brand Volvo suggests anyway) I'm looking at their DWS 06 tires now for the Alero.
I don't care for Goodyear, Michelins are very expensive and they dry rot.
I know those tire brands are not cheap, but if they last twice as long as the cheaper tires, and the cheapo tires are half the price, you still save money. (one verus two mounting and balancing fees)
Look at utilization cost instead of sticker cost
