First part is ok! "The weather could be rainy". This states that it might rain, but you are not completely sure. The next part is a little wrong. A correct way of saying it would be, "So, I would prefer it if you were to bring an umbrella". The first part "I would prefer it", means that you would like for something to happen. Adding "you were to" to the sentence states that you would like it if the person brought an umbrella with them. "If you were to" would be future tense. You could also not use a full stop, and blend it all in to one sentence, "The weather could be rainy, so I would prefer it if you were to bring an umbrella". A more natural way of saying it in a conversation would be "It might rain later", however the way you said it is grammatically fine. Another example of using "I would prefer" might be in the sentence, "I would prefer to have a hot shower before bed". Another example of "you were to" might be in the sentence, "If you were to hit me, I would be very sad".

I'd say: It might rain later. You should bring an umbrella. Saying "could" is correct too: It could rain later. You should bring an umbrella. ...But the other sentence sounds more like something I'd say :p

I would say: The weather can be rainy. So, I prefer that you bring an umbrella.