I have a 2002 Hyundai Elantra and it has been shaking since May 2015. I took it to Tires Plus and they just made it worse. If it's not the steering wheel, is it the engine?
Engine shakes a lot of times because either the engine or transmission mounts are worn, or they are not running correctly. First thing you want to do is--have that motor mounts and the transmission mounts checked. It's old enough, 14 years old, so it could be that's cracked and it's going to make it shake quite a bit. If the mounts are solid, then it's not running correctly. You can watch my video on how to fix the car that idles poorly. It could be the throttle needs cleaning.
There are a lot of things that can go wrong. Check the obvious things like tune-ups and if the spark plugs are worn up and don't work as efficiently. There are a lot of things that can go wrong, but start with checking the motor mounts. I wouldn't recommend going back to that shop. You can have a technician, such as one from YourMechanic, come to your home or office to inspect your car's shaking issue and let you know exactly what needs to be repaired.
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