Knowing how to build a fire after a rainstorm is a very useful skill: it can offer warmth, dry out your clothes, cook food, prevent predators from coming near your camp, and, most important, it builds and sustains morale if you are in a survival situation. Wet wood can be difficult to work with, but as long as the fire is hot enough, anything will burn.
Be prepared. If you are hiking or backpacking through the wilderness, always carry some form of fire starter and kindling. Some light kindling that's easy to carry and highly effective includes: cotton balls soaked in petroleum jelly (in a film canister), hand sanitizer and paper of any type. Also, it's best to carry at least two fire starters with one being waterproof matches. While hiking you can also pick up natural kindling like birch bark--one of the hottest and longest burning forms of kindling.
Find a location that's protected from wind and precipitation. Behind a large rock or under a grove of trees works well. Collect the wood around you. Don't discard pieces if they're soaked--after a long, multi-day rainstorm there will be scant dry pieces of wood around. But, remember to collect wood of varying sizes. Everything from very small twigs to downed tree trunks will help sustain your fire.
Be patient. Large, waterlogged pieces will not burn right away. Instead, start the fire with a small amount of kindling and a large amount of small twigs. Keep feeding the flames with smaller pieces until you have a few coals. Prior to this, use your knife to split strips off the larger logs--make a large pile of these. Do not put anything larger than an index-finger sized piece for the first 20 minutes. The goal is to get a large, hot pile of coals.
Multi-task. With your small kindling-based fire, begin stacking the larger pieces of wood around the perimeter of the fire pit to dry them out. Be careful not to smother the oxygen flow, and leave a space for air to flow in and around your fire easily. After 20 minutes of feeding the small fire, begin adding some of the chunks you split that have been drying around the edges.
Be flexible. You know if you've added too much too soon when the fire starts smoking heavily and hissing loudly. Immediately take off the larger pieces and add smaller branches to invigorate the flames. You will not be able to add proper logs for about 30 to 45 minutes, and the fire must be carefully guarded and monitored so as to keep the coals hot and burning.