This is the process of taking the artwork and (for the lack of a better term) separating the colors to facilitate the creation of the individual printing plates. To show by example, here's a 3 color job:
Your printer probably won't expect you to create the plates with the trim and registration marks†, but you can certainly help them out by separating the colors yourself. The best deliverable for your printer is a file with all the colors already separated onto their each layer like so:
Each of the three layers is labelled and only contains shapes with their respective colors. My answer here should help you with that process.
Pantone Solid Coated vs. Uncoated
This is largely irrelevant when it comes to printing on apparel. PMS C and PMS U are both mixed together from the same 14 base colors and they use the same formulas for each color. The coated vs. uncoated relates to how the ink will appear on coated or uncoated paper and doesn't apply to fabric.
Take a look at this picture I took of PMS 299 in the coated (left) and uncoated (right) mixing guide:
If you look closely at the mixing ratios you'll see that they're exactly the same. PMS C and U will look different on screen (and in these books), but that won't make a difference when it comes to actually mixing the ink and applying it to the fabric.
Spot color cradients can be achieved with halftones, and this is something I would advise you to work with your printer on. Halftones can be tricky and need to be set up properly (proper halftone size, PPI, frequency, angle). To give you an idea how how a screen-printed gradient would look, here's an example:
This graphic uses only two spot colors (yellow and red), but from a distance it will have the appearance of a smooth gradient.
† The registration marks are used for the initial setup of printing to properly align each of the screens. I added them to show you exactly what the printing plates could look like for that artwork. This is something that your printer would add themselves when they create the plates, you don't need to worry about including them in your artwork. Speaking as a printer, if someone included registration marks in the artwork they sent me, I would probably replace them with my own marks that I'm accustomed to using. Separating the colors into different layers is all you should worry about.
Swallow graphic courtesy of publicdomainvectors.org