Maybe someone has any clue what the problem with my transmission is. A month ago the pipe from the transmission cooler broke while driving and the car lost ATF fluid until it wasn't able to get the gears in anymore.
I noticed it had problems shifting and was able to drive for another 800 meters or so to find a good place to stop. It was losing oil and I had it towed to a local mechanic the same day. They found the problem, replaced the cooler and the radiator. Did a transmission flush and installed a new filter.
The car was shifting perfectly before and now it has problems when shifting into drive. It takes around 1 sec until the first gear engages and if you hit the gas during that second it will make the car jump.
Once the first gear is in it has no problems shifting up and down (just as smooth as before). It also has no problem with stop and go. It only happens anytime I shift from park/reverse into drive.
The reverse gear works as good as before.

Any advice is much appreciated!

 

The easiest fix is waiting the one second for it to shift into drive. You most likely damaged the pump on the transmission running it out of fluid. If it still shifts fine to the other gears I would not worry about it. A transmission shop can hook up a pressure gauge to monitor internal pressure to see how bad the damage is.