1. Wear Quality Socks
The first place to start with keeping your feet warm are with your socks. Many cyclists who suffer from cold feet have found that simply wearing decent wool socks, particularly in layers or with inserts/liners for extra warmth, goes a long way to keeping your tootsies toasty. One combination that works well is Smart Wool socks with another pair of thin, silk-based socks underneath, such as the Experia brand socks.
If you don't like the thought of wearing several thinner layers of wool or silk socks for whatever reason, you can go with one thicker pair, such as the single layer Woolie Boolie sock by DeFeet.
2. Sole Inserts
A lot of the heat lost from your feet comes straight out of the bottom, particularly if you are wearing true cycling shoes. Using a sole insert will add a layer of insulation in that area to help hold in the warmth instead of having it all blown out when the cold air blasts in.
Options include shearling insoles which are the soft, woolly stuff you see inside slippers, or the "space age" Toasty Feet insoles by Sahalie. Though these are generally pretty thin, expect a bit of added bulk inside your shoes.
A final choice is a more high-tech option like the Cozy Feet battery-powered heated insert, which uses two AA batteries per foot to deliver heat inside your shoe.
3. Use Shoe Covers
Riding with froze toes is no fun. However, consider trying shoe covers to keep your feet warm and dry. Usually between $20-$40, they are quite effective and it's the best money you can spend to keep your feet warm.
One nice choise is the Wind Dry model made by Louis Garneau, which I totally love as it breaks the wind and adds a nice layer of insulation. Other choices offer full neoprene for added protection against moisture.
Even a partial shoe cover that just goes over the front half of your shoe, such as the Pearl Izumi's Elite or Calien Toes covers will help a lot. They cover the tips of your feet, offering enough insulation to make a difference, yet leaving the back of the shoe open, keeping the air flow to allow moisture to evaporate.
4. Ditch the Clipless Shoes
If you really want your feet to stay warmer, ditch your clipless shoes. Most typical road bike shoes have vents in the bottom that allow direct air flow to the sole to keep your feet ventilated. Often this is coupled with a mesh fabric on top for circulation through the toes. In the summer, that's a great feature. In the winter, not so much. Also, some clipless bike shoes have a big metal plate in the sole that is very effective at carrying heat away from your body.
Temporarily switching out your clipless bike shoes in favor of platform pedals allows you to choose footwear like tennis shoes or boots that will keep your feet much warmer. For normal commuting or recreational riding, this is going to be a smart choice.
5. Chemical Hand Warmers in Your Shoes
The same hand warmers that hunters, sports fans and other outdoorsy-types stuff in their gloves can also go in your shoes and are remarkably effective at keeping your feet warm. You can even find smaller sizes specifically designed for keeping your feet warm. These warming agents, which are usually activated by exposure to the air, are not designed to be in direct contact with your skin. So you'd probably use them tucked in between layers of socks, especially down by your toes which are the biggest problem for most people when it comes to getting cold.
6. Loosen Your Laces
Simply by loosening your laces you can help your feet stay warmer. Several reasons why this is true. First, it allows small pockets of warm air to remain inside your shoe, in and among your feet and socks, helping insulate your feet. Second, it allows you to wear thicker socks, or an additional layer of socks if you desire, which definitely helps. Third, it provides an additional amount of physical distance between that cold, cold wind and your feet, helping reduce the degree of direct contact with the outer air temperature and your little tootsies.
7. The Unorthodox Approach - Plastic Bags on Your Feet
If you're desperate or really in a pinch, you can try a real homespun approach, and that is to stuff your feet in plastic bags, such as a bread bag or the type of bag the newspaper comes in. Some cyclists will swear this can make all the difference, but in my experience (yes, I have tried this too), there is really no air flow at all through the bag, and so sweating and the accompanying moisture build-up inside the bag is the most immediate problem.
But hey, desperate times call for desperate measures, so if your dogs are cold, and there are no other options, why not give this a try.