Hi Ethan,
I still use Circuit Wizard from my school days, its great at simulating circuits and playing around with components - you learn a lot.
I would recommend a push-to-make switch that feeds into a decade counter's input. You will see on the software (use the Voltage Levels view) that when the counter recieves a pulse, it changes from one output pin to the next. When it gets to 9, it will reset back to 0.
Unfortunately, this output must be decoded to work with a 7 segment display. For that, we use a decoder chip (also available on CW). Feed the counter into the decoder and then into the 7 segment and voila - you have wired up the first digit, 0-9.
To count to 99, you will need to copy the same circuit. This time, instead of wiring the switch to the input of the counter, you wire the 10's pin of the first counter to the input. This means that the second circuit will only recieve one pulse every 10 button presses.
To count to 999,999 using this method, you must copy the circuit four more times (six overall).
Since this is obviously a gigantic circuit with 12 IC's and hundreds of wires, I would not recommend building it. There is bound to be a wiring fault somewhere and you will spend ages trying to figure it out. Still, it is worth simulating it to get an idea of how the circuit would work.
If you want to actually build this counter, I would recommend using a multiplexer for the 7 segments. Much less hassle with wiring but a little harder to design (CW can do it though)