Don't get wet!
No I'm not being facetious, I hike through the rain forests of BC all the time, I've spent days in a row in solid rain while backpacking and setting up camp. Getting wet up here can mean death overnight even in the middle of summer, doesn't matter how hot it gets during the day, temperatures can drop to near zero overnight, if you're wet when it gets cold, then you're at serious risk of catching hypothermia, doesn't even need to get cold, you can still get hypothermia when it's warmer out, if you are wet, and there's a wind, then you are at risk of hypothermia.
What do you need to go backpacking in the rain?
So how do you stay dry? First of all, you need to have waterproof boots and gaiters, on top of that (literally, overtop of your gaiters), wear a full rain suit. Ponchos in my experience just don't cut it, you need waterproof and preferably breathable uppers and lowers, it's not just the falling rain you need to worry about, it's also the water that collects on all the undergrowth. To keep your backpack and everything inside it dry, you need a waterproof bag cover (which many packs come with nowadays) as well as a waterproof bag liner (large black garbage bags work well enough), and waterproof stuffsacks to pack your clothes and food in. You can use a poncho over top of everything to help keep your bag dry if you wish, I've yet to use a bag cover that keeps all the water out.
I feel like you often get more wet while hiking after it rains, because the underbrush is saturated with heavy drops of water on every leaf and blade of grass, which soaks into your pants and boots as you brush by. Under your rain suit, wear fast drying clothing. Polyester and nylon blends hold very little water, and shed it quickly. Cotton is awful to wear when you're trying to control moisture and stay dry, it soaks it up like a sponge and hangs onto it forever, so don't wear cotton if it's wet out.
If you only get light rain, then I'd say it's fine to hike through it in your rain-gear, just don't hike so fast that you sweat buckets under your rain suit and get soaked anyways, but if it looks like it's going to downpour like crazy, then I'd recommend pitching your tents where you are and waiting the rain out.
Something extra you need to bring is a good tarp to hang up over your cooking area. The size will depend on how many people you're camping with, it needs to be big enough for everyone to fit under and prepare their food. Under the tarp will also be where you hang up any clothes you couldn't keep dry.
How do you set up your tent in the rain?
Set up the fly first. Whether or not you can do this may depend on what type of tent you have, but I'd recommend practicing setting up your tent under your fly, even if it's a matter of simply spreading your fly out and then fumbling around with your tent underneath it like you're hiding under the covers of your bed. I have an MSR Hubba hubba, which is extremely easy to set up with just the fly, you can quickly put the fly and poles up, then bring the tent in afterwards, clip it to the poles, and spread out the corners.
How do you dry your stuff in the rain?
Wear it. As long as you're still warm, then leave your wet clothes on, your body heat will dry them out, if you get a fire going, wear it next to the fire vs. hanging it up near the fire, I can't tell you how many articles of clothing I've burnt trying to dry them by the fire. Keeping them on your body gives you an idea of how hot they're getting, so you don't get them too close to the heat and flames. Get them as dry as you can wearing them, then hang them up inside your tent, it'll be warmer in your tent than outside, which will dry them faster than hanging them out in the cold.
One thing that is imperative, is to always have a dry pair of clothes! These should be the clothes that you sleep in, keep them in a dry bag, and only wear them in your tent. In the morning, change back into the clothes that you had hung to dry, even if they're still wet, make sure that you always have at least one pair of dry clothes for the end of the day.