I'm interested in filters, flash & lighting options, composition, and post processing.
Is it possible to get really vibrant colors or should I give up on that and focus on dreary, muted, or black and white photos when the sun's not out?
You can definitely improve on the standard "cloudy day" look with some preparation at shooting time and a bit of post-processing afterwards.
As you shoot...
The two main things a sunny day gives you are contrast (bright highlights and strong shadows) and vibrant colour. So, to cheer up your cloudy shots:
Here are a couple of photos both taken on hopelessly wet, dismal days. The first has benefited from some increased contrast and saturation, the second from contrast and over-exposure. I appreciate they're both of a similar type, but considering the conditions at the time I was really pleased how they turned out.
You can have quite a dramatic effect by underexposing the background and using flash(es) to obtain correct exposure of your subject. Lighting setup should be the same that you'd use in dark for the same subject; of course, avoid spilling much light on the background, or it won't be underexposed any more.
If you're shooting film, you could choose a high saturation film, e.g. Fujichrome Velvia or Kodak Elite Chrome Extra Color.
Well, the first thing I would recommend is a polarizing filter. This really brings out the colours but it also takes some light ... about 2 stops.
I would recommend to try this first and then see if you need any more filters.
And remember for DSLRs a "circular" polarizing filter is required.