I've done lots of snow driving, in a front wheel drive car without snow tires (I do have snow tires NOW, but never used them when I was younger). Steering into the skid is not really accurate. You do want to avoid over-correcting by jerking the steering wheel the direction you think the car SHOULD go. Slow, steady movements, both steering and braking are what help you gain control. Also, understand your car - if you have traction control and ABS, your car will do a lot of the work for you. On ice, though, you can skid with ABS anyway.
My top advice for ice and snow:
1. Slow speed. Always err on the side of going slower than you think you need to. This especially applies when you are starting from a full stop or cornering. Drive at the speed at which YOU can remain in control of your vehicle. Do not try to match the speed of other cars that are going faster than you - they may have 4 wheel drive, snow tires or chains, an experienced winter driver, or just a reckless driver.
2. Increased stopping distance. Stay REALLY FAR BACK from other cars. When you are coming to an intersection where you will need to stop, begin slowing waaayyyyyy early.
3. Test your braking power - try some (gentle) braking when you don't NEED to stop to see how your car handles for that drive's road conditions.
4. Having said that, avoid braking and brake early. Try to do most of your slowing and stopping by simply taking your foot off the accelerator. If you need to brake, so it well before your stopping point. Leave plenty of space to do this. If you find you are skidding, braking can make it worse - gently accelerating gets your tires spinning again (instead of skidding) and can help you get traction.
5. Beware of hills - even small ones. ANY incline will get you in trouble. Try to anticipate when you will come to an incline and reduce your speed accordinging.
6. Trip planning - avoid hills, busy intersections, curves, and high-speed roads if possible. If you have driven the route before, you may know where the tricky spots are and can plan ahead. You may also learn when/where your city is most likely to do snow removal, sand, gravel or salt, all of which will improve the road condition.
7. Damage control - if the weather is terrible and you do not have snow tires or really good all weather tires, you will most likely skid or spin out at some point. If you are moving slowly and maintaining a large stopping distance, you should have time to mitigate the potential for damage in a skid. Steer towards curbs - hitting a curb at low speed is great for stopping an out of control skid, and likely to reduce damage. If you are coming into an intersection and find that you can't stop, lay on the horn as you go through the intersection to warn other drivers that you have lost control of your vehicle. Going slowly also ensures that even in a worst-case scenario - you hit another car - it is unlikely that you or anyone else will be injured.
8. Studless snow tires - they really, really make a huge difference in how your car handles in bad weather. Studless snow tires are NOT the same as all-season tires. And studless snow tires handle well in the rain, too (studded tires sometimes have poorer performance in rain).