There are three factors to consider when designing the puppets:

Size

The size of the puppet is largely driven by character and the context in which the puppet will be used. We have puppets ranging from about 25cms tall to life size. However, the larger the puppet gets, the more difficult it becomes to operate, particularly from the point of view of weight. Generally I think the puppets are more interesting when they are smaller than human size, sometimes considerably smaller.
It is often useful or necessary to distort the proportion of the human figure when designing a puppet. On the larger puppets I sometimes make the head smaller proportionally to the rest of the body than it would be in real life. This can help to makes the puppet look more elegant. However, the smaller puppets or non human characters, which often tend to be more of a comic caricature, may benefit from a head which is larger in proportion to the rest of the body. It is a case of deciding what works best for the character you are creating, and what you feel to be instinctively right.
I usually do several drawings before beginning a new puppet. These will include full size sketches of the head from the front and side and a full size plan of the puppet. This is particularly important if there are several puppets for one show which have to relate in scale to one another.

Facial Expression

The puppet's expression will again be dictated by its character.
I tend to sculpt expressions that are not too emotionally extreme. This seems to allow a wider range of emotion when we are using the puppet, allowing the actor/puppeteer to take the character in all directions emotionally. I usually sculpt the mouth slightly open. Again this is personal preference and not always the case, but I think it gives more possibility that the puppet is talking or perhaps just active.
I tend to sculpt from some sort of visual reference. Either a photo of someone in particular, or a drawing. Even when working from a photo, I find it useful to do a drawing, as it helps me to understand the features and structure of the face.

Construction

The most important consideration when constructing the puppet's body is where to place the joints. It is these which will give the puppet its style of movement, which is obviously a large part of its character.
I often do technical drawings for different parts of the puppets, particularly the arms and legs, in order to work out the shape of the limbs and the positions of the pivot points for joints. I use the full size plans to position the joints from the front and side to ensure that the puppet will hang in line.
Because of the structure and materials used to make the puppet and the fact that they are mostly less elastic than human flesh, there is usually a compromise between maximum movement in a joint and the best possible shape for the profile of the limbs. For example, in order for a leg joint to bend enough to allow a puppet to kneel, you must cut away above and below the back of the knee. The challenge is to position the joint so that it gives the best possible shape for the knee at each stage of its rotation and allows you to keep as much as possible of the volume of the leg. In many ways the gaps in the puppet between the sections of body and around the joints are as important as the solid parts!