Vibrating car and eight speed transmission

XL model. I have noticed that when I use the accelerator the car vibrates until it reaches a steady speed again. And I thought it was something to do with the struts or shocks and took it into our car dealership service department. They found nothing wrong with the struts, shocks, and anything related and recommended that it might be because of the road force imbalance on the tires. However, in order to "fix" it if that is the problem for the vibrating car, it is not covered in our warranty plan since it is considered a maintenance thing. However, I am worried that if I go with it it might not fix the problem and out of money. I researched online that it might be due to the U joints and the yokes, or the rear driveshaft. Another possibility may be the motor mounts and or the cross bar. So my question is, if any of the possibilities I mentioned would cause the vibration when accelerating? I do not want to go into the service center and have them laugh at my face because I just mentioned random parts that do not have anything to do with the vibrating.
My other question is the eight speed transmission. Right when shifting gears, my car jerks every time. I asked them to change the transmission fluid since it might be because of this but they told me that since my car is an eight speed transmission I should not have to change the fluid, and I also read online that I should not ever have to change it in my car's lifetime, until eight years or so. I do not really know about eight speed transmissions and not sure what to do about my car jerking when changing gear shifts if they tell me that I don't have to change the transmission fluid. What should I do? Sorry, I do not really know about cars, and my dad was the one who told me to take it into the service shop but he did not know what to do once they told us the things they told us.