Logic is designed with suitable input and output characteristics that yes, the input of a driven gate does not disturb the output of a driving gate, at least, up to a point.
There is a parameter known as 'fan-out', which specifies how many gates a driving gate is guaranteed to be able to drive. Often it's at least 10. Different logic families may have different fanouts.
For the details of how this works, you need to go into the analogue detail of how the gates are built, what the logic threshold voltages and current requirements are at the input, what voltages the outputs can reach at what current. But if you stay within one logic family, and observe the fan-out limits, timing limits, supply and ground cleanliness limits, then it all 'just works'.
If you want to mix logic families, drive logic from 'analogue' sources, drive logic from slow edges, use logic to drive other things, then you need to read the data sheets and ensure that voltage levels, current requirements, edge rates, are compatible.