Here's a few quick tips to keep in mind.
Dancing Between the Raindrops
When you look at the weather forecast on the Web you see little icons for each day indicating what to expect. And right now you probably see clouds and rain for a solid week. But that doesn't mean it is going to rain continuously for that entire time.
One key to enjoying foliage in bad weather is to figure out when it is not going to rain, and do your viewing or photo shooting then. The best tool for this is a laptop or handheld device that allows you to bring up a radar loop. This shows you where it is raining and where the rain is headed. With proper planning and flexibility you can find breaks in the weather to get out and enjoy.
For short term radar viewing, I recommend the National Weather Service Burlington radar loop. For a longer term view, try Intellicast.
In general, rain will continue moving in the direction it currently is, and that's usually west to east. There's no guarantee that new showers won't pop up, but this is at least a good estimate.
Bad weather reduces visibility due to the moisture in the air, and cloud cover can obscure mountaintops and long views. Your best bets for enjoying foliage are to "get up close and personal".
Drive on winding roads that get you near the foliage, not the highways where it is miles away.
If there is fog, stick to the lower elevations which are often under the fog layer.
Don't be afraid to get out of the car. Bring appropriate clothing and plan properly and you'll be fine -- a little water never killed anybody.
Keep Busy with Activities
Don't sit around moping during the really bad weather -- there is lots to see and do in Vermont aside from foliage. Go shopping, take in a show, dine at a nice restaurant... use your imagination. Or just relax at the hotel for a day, which is a nice treat in and of itself.
You can take superb photos of foliage during bad weather, but the number one key is to try to take the right types of photos. If you try to "force the issue" by trying to take good weather photos in bad weather, they will come out poorly.
So instead of grand vistas with lots of sky (which will look gray and drab), take photos that are complemented by wet conditions. These include close up nature scenes, rivers, waterfalls, fall displays (flowers and harvest scenes), individual leaves, water droplets on foliage, village scenes -- there's lots out there to shoot.
A few specific tips:
1. A polarizing filter is a must to cut glare off wet, reflective surfaces.
2. Bad weather means dark conditions -- use a sturdy tripod if possible, or increase your ISO (sensitivity) to ensure you have enough shutter speed.
3. The lack of sun means the sky will be much brighter than the ground (including the foliage). If you take shots with lots of sky in them your camera may auto-expose for the sky and leave you with dark foliage. Try to minimize the amount of sky in your compositions, if possible, or adjust the camera appropriately.
4. Rain on the front of your lens is a killer. It will create visible distortions in your shots. Use a lens hood to keep it off, and a lens cap until you are just ready to shoot.
5. I have sometimes gotten good shots from my car when it was raining... roll the window down and you can shoot while staying dry.
6. Bring with a towel to dry off your camera if it gets a bit wet. As long as it isn't drenched, the rain will not damage most cameras.
7. A friend holding an umbrella can be a friend indeed.
One other photography tip... it was once suggested to me to stock up on hotel shower caps - they make a great cover for your camera in the rain, and you can leave just the lens exposed without worry that the rest of the camera will get wet.
If it sounds too good to be true, its probably . . . .