Just remembering that some spray paints (Krylon at least, likely others) will be labeled for being OK to go right onto plastic, might save a step that way. Forget the primer.
Some plastics don't really paint well, Delrin and such, but I seriously doubt that's what you have. Once you've sanded it, it will probably feel dry, and be just fine. If it seems like it's got a wet or slippery feeling, then there's a chance paint won't take well but again it's unlikely. This sounds like a molded piece of plastic and not something machined.
Keep in mind shiny / matte finish when buying the actual paint, not just the color. If in doubt, go for shiny, and if needed you can finish off with a really light coat shot from a bit of a distance to give it a little bit of a matte look. This would be considered a mistake anytime you wanted something shiny (i.e. automotive) but you can do it intentionally on something like this if needed.
Go w/ numerous light coats, and don't be tempted to put the parts near something warm to dry faster, the outside can dry faster than what's underneath and fool you into thinking it's Ok to handle, or worse yet warp the parts. I've done both.
LOL 1 more thing, Home Depot / Lowes / your local big box typically has the most paint. I try to do the mom & pop Ace stores when I can but the big box normally blows them away on this, and you might want that big selection to get what you need to match the other stuff.