I am building my own Arduino and need to supply 5v and 3.3v to the project which will be plugged into a 9v DC adaptor. I am unsure what the best way to do this is and wonder if there is anything wrong with my current method.
Cascading regulators seems a silly idea in general. If the load of the lower voltage regulator is light and it is rated for the same input voltage, then just supply it directly from the input.
If its drop-out voltage permits it to reliably be supplied from the higher regulator output voltage, and it is drawing significant current then slaving it to the higher regulator may reduce the heat load of the lower voltage regulator, but clearly at the expense of the higher regulator which may well be working with a much higher drop and supplying whatever else it needs to.
Since the ESP8266 only draws full current intermittently, the LM1117 should be able to supply it when fed directly from the primary supply voltage.