I just bought this car and I love it. The only problem is that the car is shaking/vibrating at slow speeds, especially between 30-40 mph. After 40mph it drives fine. I have to point out that the streeng wheel feels fine. There is no vibration in the steering what so ever. What's really annoying is that only happens about 75% of the time. There are times when drives very smoothly. The car has fresh tires. I had the alignment done and all wheels balanced, but the problem still persists.

Any suggestions or solutions to this problems? Thanks.

You should have your engine block checked - you may be suffering from the infamous high sulfur response to Alusil/ nikasil

Check the control arm bushings.

Check the control arm bushings.
Also the engine mounts