Either by careful knife work or by hand, split your kindling lengthwise for wood pieces that burn better than whole sticks. By splitting your hardwood kindling in half lengthwise, you will expose the drier inner wood. The lower mass of these "half" sticks will also cause them to light faster than their whole counterparts.
Pines, firs, spruce, and most other needle-bearing trees are my first stop in humid weather because of their flammable pitch. For the best materials, select dead twigs from sunny areas to get your fire going quickly.
Bark is typically a protective structure meant to save a tree from a variety of things, including fire. Most barks aren't that flammable on their own. Tear, carve, or peel the wet bark off your sticks and kindling. There's often dry wood just below the surface, especially if you got your wood from standing dead vegetation.
Many attempts at fire making are doomed from the outset because the shape of the fire lay is wrong. Avoid low-slung kindling configurations. Instead, build a one-foot-tall cone of small twigs that is one foot wide. A tall tipi shape allows heat to rise efficiently through the sticks, drying them out and starting them ablaze. Don't believe me? Then listen to an engineer. A recent study by a Duke University engineer shows that the best fire shape is a cone that is as tall as it is wide.
Water-absorbing materials like paper and dead grass can drink a lot of water from the atmosphere and be very stubborn to light in humid conditions. Use more water resistant materials like wax paper or dead pine needles for your tinder.