for y in range(years):
y += 1
for m in range(month):
rain = float(input('How many inch of rainfall have fallen: '))
total_rain += rain
print("In the year", y,"There will be:", total_rain," inches of rain")
#reset variable
total_rain = 0
First of all, I corrected the indentation of "total_rain += rain" because you had it nested outside of the 2nd for loop, therefore it wouldn't actually add the rain you input each time.
Then all I did was set the total rain to 0 to reset it.
It works like a charm and is very similar to your own code.