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.
Another trick for the cold: unless it's Sweden-cold, you should be a bit chilly when you get on the bike, otherwise you will overheat rather quickly.
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 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.
[1] http://www.hucknroll.com/mens-rain-pants
[2] http://www.hucknroll.com/mens-base-layers
[3] http://www.hucknroll.com/booties-shoe-coversOtherwise, 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.
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.)
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.
For snow I suggest buying studded tires. They worked really well for me while I lived in Vienna.
Only takes one hand, after all... you still have your other hand free to use a cellphone or hold a cigarette.... :]