To make your glue, scoop some flour into a deep measuring bowl, and add water. Whisk it, and adjust your consistency by adding a little more water at a time. The end result you want is something just slightly thicker than white glue.
Once you've got that ready, get your other stuff on hand. Blow up your balloon (color doesn't matter, since you're covering it up anyway). For the paper strips, you can just cut up old newspapers or flyers you have, but I like to use plain newsprint that hasn't been printed on. It is so much easier to decorate at the end, especially if you wanted to paint your piñata a light color, like yellow, since the ink will show through unless you do a lot of painting.
Dip the strips into the glue one at a time, and press the paper against the rim of the bowl as you pull it out to squeeze off the excess.
Smooth it down on the balloon. Repeat, repeat, repeat. I do two layers of paper mache over the balloon on the first day.
The next day, tie a length of kitchen string around the balloon, and tape it into place. Remember to leave a long tail of string that you can use to hang the piñata when you are finished. Then repeat dipping and smoothing paper strips until you have covered it with at least two more layers. For older kids, or parties where there will be lots of kids who need a turn, I do even more layers to make it a little more durable. Let it dry over night again.
After it is hard, take a sharp kitchen knife and cut three sides of a hole into the piñata. Open the flap and stuff it with treats and prizes. Then close the flap and use more paper and glue to cover up the cut edge. This shouldn't take more than a couple hours to dry since you're not getting the entire project wet and gooey.
Now is the time decorate and personalize your creation. Often I use paint, but I have covered them with cotton balls (a sheep) or flowers (bouquet), and I have even added hats (Frosty the Snowman). In this case, since I was making a Raggedy Ann, I needed to add a wig of red yarn. I snipped a small hole in the top of the wig, threaded the string through, and slid it down over the piñata.
Because I used the plain newsprint, I didn't need to paint her face and I could just go straight to giving her those trademark features of hers. Sharpies work great, but nothing is wrong with using your regular markers. Or even crayons!