The only way to stop papier mache from going mildewy is to use wallpaper adhesive which contains an antifungal agent which may make it unsuitable for very young children. It`s when you use flour and water to make papier mache, then it goes off!
You can use any type of paper, brown paper is thicker so you don`t need as many layers. I suggest using between 6 and 8 layers of newspaper, preferably doing 1 or 2 layers at a time, making sure they`re completely dry before continuing.
There are many books on the subject, one I can recommend is ` Papier Mache Style` by Alex MacCormick. There are so many methods with papier mache such as layering strips of paper over a balloon ( to make a vase), or making a pulp ( which is how I made the little cats). Do give it a go!