Welcome to the Forum Mapman:
Your transformer should show 110V to 120V on one side and 24v on the other side (the high side and the low side). The vibration problem is caused by "water hammer", the water wants to continue to flow when the valve shuts it off. Its like the water hits a wall and the hydraulic pressure has to dissipate by the vibration. If you were to install a water hammer arrestor near the solenoid, you could stop the vibration.
Put a tee in the line to the solenoid and run a short (12" to 16") line up from the tee. When you turn the water back on the vertical line will hold air which will cushion the banging pressure of the water at sudden stop. The reason it works is; air can be compressed but water can't. Give it a try and post your results back.