Once, a friend of mine gave me a full set of foul weather gear for Martinmas, a day when one traditionally gives a coat to the needy. I must have seemed needy; as a naturalist working in the field, no matter the weather, I was frequently soaked. That's a bad thing, especially in the winter.
My friend's timing was perfect: It was November, and I was about to take a group of kids out on a research vessel during the first hour or so of a nor'easter. There was rain, the wind was high, and lightening stung the sea around us. I was warm and dry.
When folks ask about nature study in rainy areas, I always think of that day. If I could be comfortable in that kind of extreme weather, then anyone can get out for rainy day nature study. The gear consists of a rain jacket, rain pants, and a good rain hat or hood. These should be large enough to fit over regular cold-weather clothes. They can be very inexpensive (like a thin rain coat) or very expensive hi-tech extreme weather protection. For our purposes, since we are talking aobu t a quick nature hike in the rain, the thinner, less-expensive type of gear will do. A long rain poncho paired with high boots does well for younger kids. Once in a very unexpected torrent during a camping trip, we settled for ponchos made from large black garbage bags, a trick I learned on a schooner trip while in high school. They worked pretty well, but a long-backed Gloucester fisherman's hat would be helpful in preventing the rain from running down your back! Of course, kids love to carry umbrellas, and that's a fine way to stay dry.
Once you are prepared to stay warm and dry, a rainy-day outing is fun and different. If you go to the beach, the water might look angry and wild...or it could be calm and rippled, depending on the wind. In a rainy field, earthworms might be surfacing (they did in my yard last week, and despite the 38 degree weather, robins were pulling the unfortunate annelids out of the ground with a frenzy). Take shelter in a wooded place in the rain (unless there is lightning--take no chances!). The muddy circumference of a rainy-day pond is a perfect place to look for animal tracks. Bring plaster to make casts of any tracks you find. And this time of year, don't forget to check out the life emerging in temporary puddles or vernal ponds. Try identifying ducks while their heads are hidden under their wings.
When your hike is over, come home to a warm fire, a cup of cocoa or chowder, and a hot shower or bath. Cuddle up with a good rainy day book (Theresa has lots of good suggestions here). Or watch a rainy day classic like Noah's Ark for the little ones, or Captains Corageous for teens (lots of water).