Step 4: Climb over it (with protection)
There are many materials which can be used to protect yourself from the barbs. Some of the more popular ones include thick cardboard, anything made of leather (jackets, pants, etc.), sleeping bags, carpet, and the bodies of comrades.
The following movie clip (I would name the source, but the first two rules prohibit me) shows an example of using old carpet to quickly bypass a barbed wire fence while leaving no trace of their presence.
The old carpet / rug method is perhaps one of the best ways of getting over a barbed wire fence. The weaving on the bottom of the fabric is extremely thick and does not allow the barbs to pass through, protecting the user from any harm. To be extra safe the carpet should be folded over itself at least once to ensure no barb penetration. The pictures below show the process of cutting up an old rug and using it to protect me from the barbs while I climb over the fence. The process and all the pictures are pretty self-explanatory.
There are only two interesting things to note about the pictures. The first is that I somewhat cheated by going with the angle of the barbed wire fence instead of against it. The second is that the rug/carpet will be a much sturdier climbing surface if it is placed over an angle bracket.