There are several reasons to pressure wash a gazebo before you stain it. This applies to a first time staining and also if your gazebo has old stain on it.
Number 1: Remove Dirt
If you have a new gazebo that you have built or have had built, there may be oil from hands and fingers on the wood. In addition to that, marks such as chalk lines and pencil marks made during construction need to be removed. Loose sawdust also will be removed during pressure washing. If it has been sitting awhile without being stained it can accumulate leaves, bird droppings and dust and dirt that will inhibit the stain from adhering to the wood if it isn't pressure washed first.
Number 2: Remove Old Stain and Paint
If your gazebo has been previously stained or painted, pressure washing is a must. If you are staining it a new color you need to use a high setting on the pressure washer to remove the old stain and return to new wood so the stain will be closer to the color you saw on the stain chart.
Similarly, if the gazebo has been painted and now you want to stain it, pressure washing on a high setting will remove most or all of the old paint.
Always allow the wood to dry thoroughly before you stain any wood.