Your question is incredibly board because it can be broken into 5 distinct aspects; planetary gear sets operation, clutch and band operation, torque converter operation, control system operation and the power flow through the transmission. Further complicating the answer is that there are hundreds of different types, kinds and transmission manufacturers. For the rest of the response i will focus on the power flow through the transmission assuming that you already know how everything else works. I gave a description of how the transmission contort system works in this post and how the overrunning or sprag clutch works in this post.
I will be describing the Chrysler 3 speed Torqueflite transmission. It is the simplest of the transmissions and in school this transmission was used as an example for basic transmission operation.
The transmission has these basic components; input shaft, front clutch, reverse clutch, kick down band, low and reverse band, kick down planetary gear set, intermediate shaft, reverse planetary gears set and output shaft. The input shaft turns both clutches. The front clutch is attached to the ring gear of the kick down planetary gear set through the intermediate shaft. The reverse clutch is attached to both the sun gears of the planetary gear sets, the kick down band can also stop the same assembly. The output shaft, kick down planet carrier and reverse ring gear are all tied together. Finally the reverse planet carrier is tied to a overrunning clutch and can be held by the low reverse band.
In firs gear only the front clutch is applied. The front clutch transfers rotation from the input shaft to the intermediate shaft then to the kick down ring gear. From here it gets complicated because there is a double gear reduction that is hard to see. The kick down ring gear turns the kick down planets. The kick down planets drive the out put shaft. The kick down planets also drive the kick down sun which in turn causes the kick down planets to spin slower. The kick down sun then spins the reverse sun. The reverse sun then spins the reverse planets. The reverse planets spin on their own but the reverse planet carrier is held from turning by the overruning clutch. The rotation of the reverse planets is transferred to the reverse ring gear and to the output shaft. As i said this is very confusing because the two planetary gear sets are working together to increase the gear ratio.
To go from first gear to second gear the kick down band is applied and it stops the kick down sun. This causes two things to happen. First, because the kick down sun is held it is no longer sapping rotational speed from the kick down planets and the gear ratio goes down. Second the reverse sun is also being held. This causes the reverse planet carrier to spin backwards. The spinning backwards is allowed by the overruning clutch which can only hold it going forward. The reverse planet carries now just free wheels and contributes nothing.
To go to third gear the kick down band shuts off and the reverse clutch comes on. With both the front clutch and reverse clutch engaged at the same time the kick down ring and the kick down sun are spinning at the same rate. Because they are spinning at the same rate the kick down planets are not turning on their own axis. Instead the kick down planets are turning the kick down planet carrier and the output shaft. This whole assembly is now running at the same rate of 1:1 the lowest gear ratio of this transmission. The reverse planetary gear set free wheels just like in second gear.
In reverse the reverse clutch and low reverse band are applied. This causes the power to go from the input shaft to the reverse sun. The reverse sun will turn the reverse planets and with the reverse planet carrier being held by the band the planets will spin backwards. The reverse ring now moving backwards turns the output shaft backwards. Because the kick down ring is not held the kick down planetary gear set just free wheels.
Service manual reference.