which is a brush without any paint on it through or over top wet paint
it gives the paint a texture. I use the technique a lot just because I have a tendency to be really heavy handed with paint and if I didn't use it, I'd have paint clumps everywhere
There's no right way to hold a brush. It varies with what you're doing. I hold the brush like a pencil if I'm doing work which requires a more steady hand. If I'm making big strokes, I'll hold the brush by the bottom of the handle.
You need more than one brush. Brushes run from a size 0 to a 20. Unless you're doing some really delicate detail work, you won't need anything smaller than a 8. Unless you're doing some huge strokes, you won't need anything bigger than maybe a 12. My instructor recommends that we have a 4, 8, 10, and 12 flat or bright. I prefer bright because the bristles are shorter and it's easier for me to control. She also suggests that we have a 3 and a 6 round. Those are brushes that come to a point. Most people use them to paint a thin line. I've found using the edge of a bright to be easier.
What kind of pallet are you using? You can buy paper pallets which are nothing more than sheets of paper that are treated so the paint doesn't bleed through. I like them and you can get a pad fairly cheap.
I'm confused about the dry brush gradient though. If you're doing a gradient, the color is the most important thing, not the brush stroke. Granted you want nice clean strokes but you really shouldn't be worrying about strokes right now if you're doing gradients. It doesn't make sense that your instructor is focusing on both right now. I'd do one or the other.
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