Your puppy takes his cues from you. If you think it's fun out in the rain--or at least act like it is, he will think so too.
Spend the time with him that he needs now while he is learning and once he is grown he will be an easy care dog.
Also if you have a veranda or deck you can train him there with some newspapers but you want to give that one a head start several days in advance. Once he knows to go potty out there on sunny days he will understand about rainy days too.
Yes, puppies need consistency in order to learn what to do. Take him outside often when little, like 8 weeks, Bladders are small. Praise him highly when he does his business. As he gets older, take him out around the same times every day. They learn very fast if you keep the routine the same. I have 2 big dogs I have to walk 3 times a day. I purchased rain pants and boots with rain coat, snow pants, all I need for every kind of weather. They don't mind it.
If thats where you want him to do his business...yes, rain or shine, warm or cold. Older dogs will get finicky about going outside when they are uncomfortable with the weather but thats when they can hold it all day or longer. Puppies do not have that control. Its miserable for you and maybe him for a little while but they cope much better than we do. It will definitely be worth the effort to have him tell you when he needs to go out in a few months rather than have him look out and see it raining and decide to go somewhere in the house instead! IMHO
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