Thank you. where do I connect the output of the monostable?
You use it for gating the astable multivibrator, see my post #10 and below.
does this monostable act as the entire cycle of the flashing lights, or does it simply make the lights stay on for say 1 second, then repeat with the reset?
A monostable multivibrator is normally off. When triggered, it will be on for the time set, then turn off again. It will not automatically repeat the cycle - that is what the astable multivibrator does.
the Gating website does not really help with gating the monostable to the astable, just mentions astable to astable.
It is irrelevant whether the gating signal comes from an astable or a monostable multivibrator. In both cases the second timer (#2, the astable one) shall be active during the high period of the gating signal only. Simply use the monostable multivibrator instead of an astable one to create the gating signal you need.
With the information you've been given you should at least be able to create a
block diagram of the circuit., i.e. a diagram showing the functional units and their interaction without going into the details. E.g. show a monostable multivibrator as a block labeled accordingly without drawing the internal circuit (no Rs, no Cs, no 555s etc.).
Show us how you think your system looks like on a high level in a block diagram. The next step will be to refine this block diagram into a real schematic.
While you're at it you can already start creating an astable multivibrator (short mv) and a monostable mv as (separate) simulation models in LTSPICE so you can find the required component values test both of the mvs individually,
The next step is then bringing the two models together and adding the gating function.