When you are cycling in the suburbs, you really cannot skip checking the weather conditions! Sometimes I still forget, and sometimes I really regret it.
The weather conditions I try to avoid are: thunder, gusty winds, and ice/snow. Rain is not a dealbreaker, but anything beyond light drizzle is annoying with groceries. Also, in the winter rain is extremely cold! Rainy 34F can be more problematic than dry 0F.
You might find this surprising, but dry cold is no problem! Proper clothing is essential, but you can be very comfortable. I have personally cycled at 0F and I am sure you can go below as well. Getting stranded in the cold probably stinks (never happened to me). I also carry tire slime and a pump, and a cell phone for emergencies.
It should be obvious why thunder and ice/snow are risky. But when you’re cycling in the suburbs, even light wind gusts can be tricky because you’re probably riding near to traffic passing you at high speed without much regard for your safety or the weather conditions. And since sudden crosswind gusts can push you sideways into traffic with no warning, I think it’s smarter to postpone your trip in these conditions.