Just the verb "padać" can be enough, but it can mean both that it's raining and that it's snowing. If someone says "O, pada!", you just assume the most logical thing at the given time of year. In winter you'd expect it to mean "Oh, it's snowing!" and in other parts of the year "Oh, it's raining!".
Or you can just be more precise and say "Pada deszcz"/"Deszcz pada" for "It's raining" and "Pada śnieg/Śnieg pada" for "It's snowing".
In theory there's also hail (grad), but that's too unusual, it should be specified ;)