First step is to check your engine mounts and transmission mounts for free play.
Engine mount free play can be check by using your han d to push the engine front and back, left and right. Do this when is engine is cold, you don't have to start the engine. If the engine can be moved, then the engine mounts may be worn out, or loose.
Since the transmission is mounted below the engine, there is no easy to check it by the above the method, but you can always do a visual inspection, and look for signs of problem. It is worth to bring the car to the repair shop to have mechanic to check the problems.