My car's headlight and dome light flickers when the car is started, but when the car is already moving light is working fine. Other lights works fine, though I wasn't able to check the tail lights and the light in the trunk. Before I start my car, I ensure all lights and A/C are turned-off.
This is very common, and not a problem.
When you are starting your car, the load on the battery is huge compared to when the engine is running, as the starter motor is having to turn the crankshaft and push the pistons while the oil is cold and thick, and additionally compressing air into the cylinders, pumping fuel and causing a spark, while also driving belts, fans, alternator etc.
As soon as the engine is running, the alternator begins to top the battery back up, but to give an indication of how much the battery is loaded during start up, if your alternator fails, you can drain your battery in only a few attempts, especially in cold weather.
It is good practice to have aircon, lights etc. Turned off while starting the car, for this very reason.