Here's my attempt to make a truly thorough answer to the question of cold-weather clothing. Honestly many chapters of books have been written about this subject, so it is hard to give a specific answer for a specific case.
First - Generally agreed upon principles:
- There are three things that impact hypothermia - wind, wet, and cold. Given the right circumstances, only one of those three is necessary to seriously injure you, though usually two or three get dangerous. For example, it might not be windy, but it is cold and you are wet - this is a recipe for injury. It is certainly possible to become hypothermic on a cool summer night in the Rockies if it is also very windy or you are wet.
- There are two primary ways to get wet - precipitation (at these temperatures, probably snow), and perspiration (sweat). These things should be avoided or protected against if at all possible.
- The best way to manage your warmth safely (without sweating, but being protected from wind, wet, and cold) is layering. REI has lots of examples and more layering discussion at Layering Basics. In short, the system involves roughly three layers. The first is a next-to-skin layer that wicks moisture from your body while keeping you somewhat warm. The next layer is an insulating layer that creates dead air space to keep you warm (it also continues the moisture wicking process). Wool, fleece, and down are examples of this layer. The last layer is a shell that protects from wind and wet. Examples would be a wind shirt or waterproof-breathable jacket (of which Gore-Tex is only one example). The layers can be taken off or added as conditions dictate, thus keeping you warm, dry, and optimally protected.
- "Cotton kills" - This is an exaggeration, but it comes from the reality that when cotton becomes wet it no longer insulates and takes a long time to dry out. This means that cotton should be used with caution and in the right circumstances. Don't use cotton as a baselayer in cold weather or where it might get wet and put you in danger. I use cotton shirts when taking certain summer trips because it's more comfortable for me, but I avoid situations where it might harm me by not drying or by cooling me too much.
- Protect yourself from frostbite. This means that your face should be covered as much as is reasonable. If extremities start feeling numb, deal with the problem before it becomes serious. Exposed skin can get frostbite very quickly in cold, windy circumstances (see: Wind Chill Temperature Table).
Next - consider the circumstances:
Other than the temperature (we've already established that it is very cold), we need to consider the other elements of weather such as sun, wind, rain, and snow. On sunny trips you will need something to protect you from sunburn, particularly if there is snow on the ground (the snow reflects sun back into your face from a new angle), and particularly if you are at higher altitudes (atmosphere is less dense and thus blocks fewer rays). When there is less sun, the impact is less (but not gone - don't stop protecting yourself). Wind adds another element to consider (see: Wind Chill Glossary). Rain and snow contribute to being wet, which is one of the pillars of hypothermia (from above). If there is even a distant chance of rain or snow, it would be foolish to not carry waterproof protection.
Nature of Trip:
If the trip is a gentle stroll through high meadows, close to safety, there is more wiggle room when it comes to gear. If the trip is a 25 mile boulder-bash far from safety (perhaps from a hut or deep in the wilderness), you can't afford to cut corners with your clothing. This question specifically seems like an intermediate, which would require some careful consideration.
Skill, attitude, and experience of hiker:
Some hikers have many particular systems designed to be used in specific circumstances. Those systems won't work for everyone, so we have to be careful in making specific recommendations. As a guide, for example, I would instruct my students not to bring a single scrap of cotton with them on an intermediate day hike (aside from a bandanna, perhaps). If I were doing the same day hike, I might wear mostly cotton - it depends on the circumstance.
Finally - Some recommendations:
Baselayer: Wear a synthetic or wool baselayer on top and bottom (examples: Capilene from www.patagonia.com and baselayers from www.smartwool.com)
Insulation: Wear (or carry, if the effort will be high) a mid-weight insulating layer such as a fleece top and bottom (examples: Aspiring Hood from www.icebreaker.com and Vermillion Thermal Jacket from www.golite.com)
Standing insulation: When you stop (particularly if you will be stopping long-term) you will probably need more insulation to keep you toasty. A down jacket is very common for this. Lower fill ratings (500-650) will be heavier and bulkier for their warmth but are also cheaper (example: Beartooth jacket from www.golite.com). Higher fill ratings (700-850) will be lighter and more packable for their warmth (example: Bitterroot jacket from www.golite.com). I personally love my down to have a hood on it (it keeps me warmer at night, too), but that is up to preference. Most people keep a down in the top of their pack, ready to be taken out when stopped.
Shell: If there is zero chance of getting wet from rain/snow/etc, there is little need for more than a wind jacket (example: Trail Wind Hoody from www.marmot.com). If there may be precipitation, use a waterproof-breathable shell. There are thousands of these on the market, so there are lots of possibilities there. A hood is very nice to have for this layer, both for your comfort and safety. If you aren't into hoods, that's okay - just make sure you have total coverage for your head and neck.
Hat: Wear one. A balaclava (a type of hat that covers your head and neck while leaving a hole for the necessary parts of your face) is useful for very cold environments, and can protect your face from frostbite. Something with a brim can help shield you from sun, also (REI sells a selection of balaclavas).
Gloves: Dependant on circumstances. I know people who make do in very cold environments with simple liner gloves (example: WindWeight Glove from www.blackdiamondequipment.com), at least when all they are doing is hiking in cold places, but I use a much warmer glove (such as the Ambit or Alti Glove from www.outdoorresearch.com) almost all the time (I have cold hands). Keep in mind that a big glove reduces your dexterity, so it is a good idea to have two pairs - one liner set and one big warm set. Mittens are a good option for really cold situations.
Socks: Again, dependant on circumstances. I wear thick wool socks most of the time (www.smartwool.com), but I occaisonally wear a set of synthetic liner socks if the conditions are wet, because they wick more and prevent blisters better (for me, at least).
Gaiters: These wrap around the top of your boot/shoe and go up your ankle. They can be found in a variety of sizes and styles. I only wear them in deep snow, but some people wear them all year long to protect their ankles from wind, cold, or scrapes. www.outdoorresearch.com sells a large variety of them.
Experiment with your layers.
On a day hike, you don't have very much risk, so experiment with what you have. Take a backpack that contains many layers and try them out. It will give you a better perspective for when you really need them.
If you start a hike feeling chilly, you're in the sweet spot. If you start warm, you'll get too hot and sweat (which is bad).
Pay attention to your body.
Are you freezing with all your layers on? You should probably turn around. Are you sweating like a pig but don't want to ruin your momentum? Don't risk it - take off a layer. Did you barely survive the last trip you took? Assess your system and adapt it for the future.
Go on trips with other people.
Ask and observe. Find out what other people use and what works for them. There are ten million opinions on what really works and what doesn't - try to separate the wheat from the chaff.
There. I'm sure there are things I missed, but I tried to keep it concise but thorough. Feel free to add or modify.