(This ... is going to be a long post

But it will likely be very comprehensive.)
Sprays that tend to be used:
Testors Dullcote
ZM (Zoukeimura) Finishing Powder Spray (UV cut) (available from Volks/Volks USA)
Mr. Super Clear (MSC) -- flat, regular and UV cut (available from Volks/Volks USA, if in the US you can find it cheaper elsewhere, though ... Volks will rape you with UPS shipping XD; )
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Because I've used all three, I'll give you a quick comparison. Please note this is based on my own experiences and my own preferences, different people will have different opinions XD
Testors: In my experience I found that testor's spray tends to attract more dirt than the other two, so over time you may find that your finishing coat actually needs to be cleaned. It also has a noticeably yellow tint--though this could be a good thing if you have a hybrid that you are trying to match resin tones on.
I have had less problems with temperature/humidity related issues with testors than the other two.
ZM Spray: ZM spray has a middling amount of tooth (this will effect how well pastels will adhere to the surface), I don't actually remember how Testor's compares to it, but I know that MSC has much more. It'll take you a few coats to build up to brighter or darker shades of a color. I has no noticeable tint to it at all.
I have had less problems with temperature/humidity related issues before, but less than with MSC. (These kind of issues with spray sealants generally involve either taking on a foggy white cast, or otherwise not setting up properly ....)
MSC: This has become my personal favorite. It is much easier to do pastel work on MSC because it has greater tooth. You'll need less layers of sealant and the color you get brushing on will be much truer to the tone of the pastel itself (ZM spray tends to make it a bit lighter because it doesn't stick as well). Although I've not notice much tinting at all, other customizers have noticed a slight yellow cast to MSC (though not as much as testor's).
I have had more problems with temperature/humidity related issues with MSC than the other two sealants, it appears to be more sensitive to these factors. So you'll have to be careful. It is also more harsh in terms of fumes than the other two. Of course, it is generally recommended that you wear a mask or respirator when spraying any sort of sealant or paint ....
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Sealants out of the way, let's move on to materials actually used to paint a doll.
Airbrushing: I've ... not done a lot of airbrushing personally, mainly because all attempts I did at it failed. So if you want to try this you'll have to ask someone else 8D; Most likely you won't be doing fine details with an airbrush, just your blushing ... so you don't necessarily need to go buy the most expensive brush you can find. If you plan to give this a try, do a little research and keep in mind what you'll need it to do.
Pastels: Pastels can look just as nice as airbrushing if done right (and it'll take you some practice). You want to avoid oil pastels at all costs. The ones you are looking for are called
soft pastels.
You can use cheap or expensive ones, the expensive ones tend to come off the stick smoother and have more range of colors. If you use cheap pastels make sure they are ground as fine as you can get them. In the past I've often scraped at a pastel stick with a knife blade to produce a powder.
(As a note, I have had slight issues with red and black cheap pastels staining, so if you plan to use these colors be sure to coat extra well on your base coat. I'd actually recommend doing this if you plan to use black, red, or any other highly saturated color period, no matter what materials you are using.)
The brands I use personally are
Rembrandt and
Schminke. These are both a tad more expensive, though (particularly Schminke).
You'll want to choose natural colors for your base. Reds, pinks, oranges, and browns. Also consider a good light blue, because it actually is a tone in human skin. If you plan to paint a doll with makeup or a dramatic/fantasy faceup then you may want to consider other colors as well.
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Detailing:
There are three options I've tried personally for detailing. Watercolor Pencils, acrylic paints, and watercolor paints. I'll talk about them all.
Watercolor Pencils: This is a good option for a beginner. Especially if you don't mind the grainy texture that the pencil will leave behind (I hated this so I moved on to brushwork). It is easier for most people to control and so long as you keep the pencil sharp you can get fine lines. They are also relatively easy to clean if you make a mistake, as they are water-soluble (this means that they thin/dissolve in water).
As a note, some people use colored pencils, but I do not recommend these. In particular I have seen Prismacolor Pencils mentioned--but these pencils are wax-based, and wax is an oily substance.
Brushwork: The alternative to pencils is to paint on details with a brush. You will need a small brush to do this and the ability to control it, which will come with lots and lots of practice. There is a learning curve to this, though ... so unless you paint a lot already don't expect to become an expert at it quickly.
Watercolor Paint: Tube or pan watercolor will work, though tube colors are easier to get working consistency with. Watercolor is very easy to thin with water to get different levels of transparency for working with faceups. It also has a thinner consistency than acrylic paints and so it can be easier to work with.
On the down side, if you compare watercolor and acrylic brushwork ... acrylic is always going to have sharper details, watercolor will always have a tendency to go fuzzy or to run/bleed a bit. People who paint the details on their dolls are often looking for this.
Acrylic Paints: Generally you will need to thin these with a paint thinner (you can thin with water, but thinner will change the color less; water has a tendency to make the color lighter as well as more transparent). I've personally not had much luck with thinners, so I don't have any to recommend at the moment--I will be trying out the Zoukeimura thinner soon on recommendation, so if it turns out to be as amazing for me as others I can write again about it.
Acrylic paints can be softened with a bit of thinner or water if it goes on too sharp for tastes, and thinned to be less opaque. Golden paints or Liquitex are the generally recommended brands, personally I tend towards Golden's Fluid Acrylics.
(As a general rule of thumb with paints, student grade paints are lower quality, artists grade are higher quality ... student grade paints are "watered down," so to speak, with more binder than the materials used to give the paint color--so basically you get less saturated, less rich colors. You can use this to decide what grade of paints you want to look at. People have used cheap acrylics to as much success as more expensive ones. If you plan to commission you definitely want more expensive, quality materials; but for personal face-ups you may find that cheap materials work just fine.)
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Just a few other things that are helpful for faceups:
Wide variety of larger brushes: I find that round brushes work best for me in faceups in general, though you may want a flat or two as well. If you plan to use brushes to apply your pastels you will want a wide variety of sizes from smaller brushes for more detail areas (corners of eyes, lips) to wider brushes for large areas (cheeks). I find brushes tend to give a softer touch to blushing.
Cotton Balls: Cheaper alternative to brushes, but I also find you sometimes get a bit of cotton that comes off in your faceups.
Q-tips: Q-tips tend to lend a harder touch to blushing, and I find it can be very good for where you want a darker spot of color. I use these in conjunction with brushes. (In theory the pointed-end Q-tips would be better for details, but I can't ever find these to give it a try.)
Erasers: Some people have suggested using magic erasers previously. I would caution against this because they have a slightly abrasive texture to them that can take off your sealant. Wet they will certainly take off your sealant.
What I would recommend instead are white erasers, you can get these really cheap at an art store or kneaded erasers. I personally use both. White erasers are good to use when you want to take everything off, kneaded erasers can be used when you get something a little too dark and just want to take off a little color. You can squish it to a shape that will cover the area and dab at it till you get enough removed.
Gloss: I've heard excellent things about Tamiya glosses, which Volks sells ... but have never tried it personally. What I use is Liquitex gloss medium, which works fine.
(As a note, if you'd like a more matte gloss, the Liquitex matte medium, which I have mostly to lighten colors in my acrylics, has a slight shine to it. It can be used much like gloss if you want something not so shiny.)
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Additional note, relevant to the original question:
So far as I know people use the same materials to paint vinyl and resin. Vinyl is much more prone to staining, however. So I recommend you put a thicker base coat down (this will give you a surface to paint on and also serves as a buffer between the vinyl and the actual paints). I would also caution using very dark, highly saturated colors. Be careful if you use black or very dark reds--these stain surfaces the most easily.
Hope this is helpful ^^
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