Input: variable voltage; Output: 3 LEDs switching depending on voltage.

the way that I would do this is a few comparator circuits using op-amps, you would need a few different set voltage supplies (4V, 6V, 8V) or just an 8 with voltage dividers.
or you could use analog to digital converters and do it that way but that would be more complicated with only 3 outputs. Im talking encoders/decoders, probably a multiplexer in there, complicated.

using the comparator method would grant you a lot of fine tuning if you want/need to using pots

 

Look at the LM3914. This has 10 comparators and 10 outputs to measure voltage.
You could neglect the first three outputs and drive a led from 4 and 5 (use a diode from each output to stop interaction) and so on to the top. The top led could be a flashing type.

The chip has a voltage reference.