I prefer to use a combination of overlay and multiply. Overlay tends to make things very saturated and light, multiply makes things often darker. Just duplicate the colorlayer, set one to overlay and one to multiply and play around with the opacity. I also sometimes use Hard Light to get some more unexpected effects. Whats also great fun is using Gradient Maps. Go to Layer > New Adjustment Layer > Gradient Map, select a gradient and play around with the type of layermode again. Sometimes just messing around with these things get you nice color effects. But remember to always color some on top, otherwise the spectrum of colors will be too limited and the work will seem boring.
Multiply is mostly for if you don't have a greyscale, only black and white. Only line art and things like that. It mostly colors only the white areas but doesn't color any black areas. So if its a grey color, it will be likely the painting color like yellow will be desaturated.
Pretty much yep, saturation is like that.
If your a traditional painter, you can also call the Greyscale an underpainting, and the next layer like saturation is like glazing or a thinned colored. But you know, just something to connect to.
I personally prefer starting from colour at the start as I find the colour more vibrant and alive but there are awesome people out there using the black and white to colour technique too, so it's personal preference.