My son and I routinely tromp out into the woods behind my house to have an "adventure". I bring the backpack that I keep that's sort of a hybrid Every Day Kit/Car Kit/Personal Survival Kit/Bugout Bag/Diaper Bag/Equipped Dad Bag.

In that bag, you'll find a wide range of devices that are useful for fire starting.

In the last "adventure" it was raining, it had been raining, and was going to keep raining for hours more. Still, as is our custom, I was going to build a small fire to warm up. We always get one going, in all weather, including heavy rain.

First, and most importantly, is a huge supply of decent tinder and kindling. You need much more than you think. Especially when it's wet out, you're going to need it just to get things going. Next, into the ever-popular plastic ziplock bag or a trash bag or something to keep the rain off, I shave dried sticks I found up high on trees and such with my knife, so they stay dry until ready to light.

Another key to success is a means of protecting your little fire as you get it going. We usually build a fire ring that's fairly deep out of some stones, if that's not an option, I've found that a small fire pit with a bit of a vent trench is a pretty reliable way to keep the small fire started. If it's really raining hard, I build a small lean-to style canopy (tiny, just a few sticks and some greens over the windward side of the fire to keep the bulk of the rain off, this really helps, and once the fire's really going, it burns off.
I also make sure that I'm patient. Sometimes I'll sit for 20 minutes feeding tiny strips of wood into a tiny fire that's just sputtering along until I get enough of a thermal mass to really light stuff bigger than a thin strip of wood. I've found that even with a Trioxane tab, you can't really build a decent fire that will last until you have a solid base of coals to heat and ignite things.

Finally, I always surround my fire with more firewood to dry it out some. This is expecially important in a smaller fire, since you will need a fairly steady supply of small wood to keep it going.