You will need:
Newspaper
Flour
Water
Bowl to mix the paste
Fork
Vaseline

If you know what you are going to make it is sometimes helpful to make a loose sketch of what you want your finished project to look like. If you just want to go with the flow and just start making a shape and then see what it turns into, you will find paper mache to be a great medium for that.

To get started cut or tear your newspaper into strips about one-two inches wide. Length isn't a big factor at this point, if you find you need shorter strips just tear at the desired length.

To make your glue/paste mix a one cup flour with one and a half cup water. Stir until there are no lumps. To thin out the paste slowly add in more water a little at a time and mix it in completely before adding more. If you make the paste to thin just add in a little bit more flour. The paste should be thin enough that when you scoop some out with a fork it runs off, but thick enough that it takes more then a second or two to completely run off of the fork.
Most paper mache projects are started using a mold. Some common molds are

Balloons
Balloons are an inexpensive way to create different sized round or oval shapes. Just blow up the balloon to the size you want and add your newspaper strips. After you have added at least four layers and allowed ample drying time you can pop the balloon then continue on with your project. If your paper mache project will not be fully closed but is more then half way up the sides I recommend using a balloon.

Cups and Bowls
If you choose to start your project with a cup or bowl to form a shape, you will first need to cover the bowl with Vaseline. The Vaseline will allow you remove your paper mache after it has dried. I still recommend at least adding at least four layers and allowing it to fully dry before removing the bowl.

Empty bottles
Empty bottles are great because you typically have more freedom in shaping your paper mache. I say this because empty bottles often do not need to be removed from the paper mache project. Of course if you do not want the bottle inside your project you can just cover it in Vaseline before applying the paper mache.

Once you have choosen you choice of mold you will want to dip your newspaper into the paste. I recommend pulling the newspaper through two fingers to remove excess paste. Lay the strips flat on the mold. Then lay the next row slightly over lapping the first. Continue until you have covered the area you want. Repeat until you have three to four layers. Allow your project to dry over night before removing your mold. You can then add more layers allow to dry after every three to four layers. Continue until you are satisfied with the thickness.

If you wish to add a different element to your project, say a handle, cut the shape you want out of cardboard making it smaller. Attach it by adding layers as you normally would around the new piece and base object.