Wear the right clothes and bring a set of clothes to change into. :)
I bike to work in Ireland (150+ rain days a year - yay). Rain clothes takes care of that - just make sure they aren't too heavy.
When I was younger I worked as a mailman in Sweden for three winters, using a bike for 2-3h a day, every day. You don't even need a lot of clothes for that - a fleece over a set of thermal underwear will do you down to minus 10 C or so, after that you want a thicker jacket on top. Heat from the exercise keep you warm enough anyways.
For both rain and snow - get a good pair of gloves (hands are the only part that really sucks when wet/cold). For the cold, get a decent set of thermal underwear.
Fenders, and don't skimp on gear. I rode every day for two years straight in Chicago, back a few years ago before the really mild winters. I rode into the negative teen temperatures whether it was snowing or not.
I switched to clipless and picked up some cheap but good Northwave boots from Wiggle in the UK. I got some wind & weather resistant knickers from Swerve. Picked up a bunch of of merino wool base layers and a thicker wool zip up, under a goretex shell zip up. Many pairs of wool socks as well. Pearl Izumi lobster gloves were great. Pick up a balaclava or some other head/face covering, a cheap pair of anti-fog goggles, and you're good.
It sounds like a lot to buy, and maybe it sounds expensive, but considering the use I got out of all of it, I didn't blink. Plus it was more fun, and in my case it's roughly the same time to ride as it is on a perfect day timing the bus/train combo I needed.
1) Wear rain jacket and pants. Breathable waterproof gear is best; doesn't have to be gore-tex. Expect to spend at least $150 on a set, though. This is what I do. Bonus, keeps you warmer in the snow. Minus, makes you sweat if it's a warm rain.
2) Ride and just get wet. Change when you get home/to work.
3) Take the bus. As the Bike Snob NYC says, IIRTTB is the hipster way.
For rain in the spring and summer I just get wet. We have a locker room with showers at my work and I have a waterproof bag, so I just bring a change of clothes.
For snow and cold rain there are water proof rain pants [1], base layers [2], and shoe covers [3]. If you stay dry then staying warm isn't a big problem because the exercise will warm you up. I live in Colorado, and when it's snowing I put a water proof layer over my normal cycling clothes, and it's usually warm enough.
Heh... they say that there's no such thing as bad weather, only bad gear! With that said, it really depends on where you live and what route you take to work. In my city (one of the colder parts of Canada), I can ride a road bike in the winter since my route takes me into downtown and the streets are usually well-plowed when it snows.
Otherwise, a mountain bike with some studded snow tires can get you by - it's pretty invigorating to make tracks in fresh powder in the mornings!
Generally, the heat your body produces while cycling is enough to maintain your core temperature. Just keep your extremities well-insulated and your body as dry as possible.
Warm rain doesn't matter, except that it affects your stopping distance so you need to brake earlier. Rain gear is mostly pointless in warm weather because it reduces ventilation and leads to getting wet from sweat rather than from rain.
For colder rain, I use a Showers Pass jacket, and I ride a bike with fenders.
For snow and ice, I ride my mountain bike with studded tires. (The studs aren't really needed for snow but they help a lot with ice.) And I switch from clipless pedals to flat pedals so I can get a foot out faster to catch falls. (I fall about once per year in the dry, versus several times per ride in bad snow.)
Get a pair of fenders -- at least a rear fender (prevents 'skunk stripe' effect) but go for a full set if you can. There are detachable ones that will fit almost any bike, but the ones that bolt on to frame mounts provide more coverage and are less likely to move out of place.
Fenders are incredibly effective at keeping groundwater off of you. In my experience this is at least as important as wearing rain gear.
Make sure you have proper mud-guards - they make a huge difference if it is wet. Proper cycle clothing makes a huge difference as well - lightweight waterproof trousers, high visibility cycle jacket with a nice long back and decent gloves.
Although sort of a cheap way out, my favorite solution is to drive once a week (I choose Friday) and drop off four days of work clothes. That way I don't have to bike with any bags at all, and can get as wet as I want during the ride and just shower when I get to work.