I'm trying to say:
I don't like the weather today because it is ____. (rainy / raining)
I have to carry an umbrella for ____ (rainy / raining) weather.
Should I use rainy or raining?
Also, what is the common rule for using these words?
To describe what is actually happening right now, you use the verb form:
It is raining.
To describe the sort of day it is, you use the adjective form:
Today is a rainy day.
In your first sentence, either rainy or raining could fit, depending on what you actually want to say; "... because it is raining" indicates that water is physically falling from the sky right now, while "because it is rainy" indicates that it is the sort of day where rain is extremely likely to happen, but doesn't necessarily mean that rain is falling as you speak.
In the second sentence, only rainy fits because it is describing a quality of the weather, not an action.