Part of the fluid does stay inside the transmission, mainly in the torque converter. It seems very odd to me that it does not have a filter inside. i am a mechanic and i service transmissions all the time and i have never seen one that doesn’t have a filter. to get to most filters you just have to unbolt the pan and drop it down. just be sure to clean the pan well and also get all the metal flakes off the magnet inside.
When your first starting the car up after initial service, shift back and forth between drive and reverse several times to get the fluid pumping through, if you do not feel it shift into gear at all after several times switching between forward and reverse then check your fluid.
Other than that they are really pretty straight forward to do. Good luck