Most of the villages around Neuchatel have a soccer teams, and it would be a great way to get integrated. They are quite earnest mostly with off season training etc.
Search google for "FC village_name" (FC=football club) eg http://www.fc-auvernier.ch/5301.html
I can also give you the name of someone who played a lot of club football in the area and might be able to give you some more precise tips.
For pick-up games, less sure, but go to any park at the weekend and there will be kicking the round ball around.