How do I get my 555's output to cycle through different paths with just one button?
Hopefully I am not breaking any rules here, but unfortunately, I do not have the rep to just add a comment above, so this answer basically just adds to HandyHowie's suggestion (and also saves me the time of drawing out the entire schematic).

That is exactly what I would have done, except:
Using the circuit he supplied, remove the 555 timer connected to pin 14 of the Decade Counter and replace it with this circuit:

Tactile Clock Pulse

Pressing the momentary switch will advance the counter, rather than the timer.

EDIT: For what it's worth, here is a schematic of what I think you are looking for. I just used a block to represent your existing astable circuit. This will give you 8 steps sequentially activated with a button press. If you want to utilize all 10 available steps, just add the Resistor/Diode series to Pins 9 &amp; 11, and ground Pin 15. Of course you can also add versatility with potentiometers in each R/D series to be able to adjust the tone of each step, too.

Sequential Tone Generator