Vinyl and plastic can be tricky to get paints to truly adhere to; if you're looking for an overall recolor of the doll to a fantasy tone, consider dye instead. You won't have any impact on the mobility of the doll, and as it penetrates the outer surface of the doll, you'll see much less wear and tear over time. Dyeing for recoloring has been used regularly in collector communities, particularly with the Dollfie Dream line from Volks and the Obitsu line from Parabox.
You'll need a dye designed for synthetics, also known as a disperse dye; check the labeling to ensure the dye specifies it's for synthetic fabrics, as union dyes ("all purpose," and not great at any of them) and dyes for natural fabrics won't work. Commonly available brands in US stores are Rit's "Dyemore" and Jacquard's "iDye Poly," as well as a large number available online. You'll also need a dedicated pot large enough to submerge the doll (or component parts if you're able to do any disassembly) without it touching the bottom or sides; an old or thrifted pot is great for this, as you should not use it for dye only, never for food. Additionally, you'll want a large bowl of cold water for cooling and rinsing the doll afterward--again, make this a dedicated item.
Keep in mind, too, that you'll be layering over the existing color, rather than replacing it with an opaque layer. For both the purposes of practicing the technique, and of getting the color correct, you might want to pick up a couple of similar dolls to sacrifice to testing, before dyeing the "good" doll.
You'll need to measure and mix your dye to work toward the color you want (more dye for a darker color, less for a lighter one), use hot (boiling) water, and use something to lower the doll in that won't hold the dye (I've used an aquarium net for smaller parts when dyeing different types of dolls, and fishing line or similar non-absorbent strings could work for larger ones--don't make my mistake and use cotton, as it will leave a mark behind where it held the dye longer). For both the test parts and the final doll, you'll dip them into the dye for a period of time (worked out in the test), avoiding the edges (which will cause blank spots, melting, and scorching), then transfer the parts to the cold water bowl and rub them gently to remove residual surface dye. After removing the excess, you can lay the doll or parts on a towel or hang to air-dry.
Forum user elianti has a very detailed write-up of their experience using dye for Volks and Parabox parts at the Doll Dreaming forum that can give you additional details on recoloring dolls with dye.