Is Transmission or torque converterv has problem?

Transmission or torque converter has some serious problems. When pressing on the gas pedal from low speeds there is barely any acceleration in 1st and 2nd gear under 2,000 rpms, but when 3rd gear goes in car accelerates more than desired most of the time. This is most apparent when car is cold, but never goes away completely. This really bothers me as this usually happens during turns when you slow down and when you want to accelerate you don't know how hard to press on the pedal, if you press too much car jumps ahead, if not enough you barely move. Another problem is that when coming to a complete stop when transmission downshifts you can hear a faint thump of every downshift somewhere under the vehicle and a little surge forward. What is the problem? Any solution?