I mostly paint for blood bowl so tend to buy colours based on what my next team needs, then go with an eye towards what might have broader utility, so can’t really recommend a basic spread. That being said, there are some basics that get used regardless of team.

Cavalry brown, leather brown, beige brown, and pale sand would cover a wide variety for skin, clothes, ground, etc. pale sand is amazing for bone and the like. Cavalry brown ends up in so many of my colour schemes, but that’s maybe a personal preference thing.

Black grey and white grey are going to cover 90% of your black/white needs. Pure black is nice but not strictly necessary, and pure white is overkill most of the time.

Then just get primary and secondary colours as you need them, one darker one lighter and you’ll have plenty of options when mixing. I recommend this over fewer paints and mixing primaries to get secondaries because the more you mix the less vibrant and saturated your colours will be.

If you’re getting metalics, it is worth it to splurge on vallejo metal colour airbrush paint. It’s just that much better. I have dull aluminium and magnesium which I can mix and match to get a range of regular metal looks. The texture is perfect out of the bottle without thinning. Every time I reach for my game color brassy brass or polished gold I struggle to thin it well and regret the couple dollars I saved.

Where possible get model colour. Game colour is usually fine but it’s sometimes hit or miss. That being said I’ve gotten an unreasonable amount of mileage out of turquoise and jade green game color + a tiny bit of black grey for fantasy skin.

I paint in a layered, build up from dark to light style and use a lot of blending though so ymmv if you want a different style

Honestly that heavily depends on what you’re painting. I’d say black, white, dark blue, deep green, a mustaurdy yellow or orange, any red and a brown go a long way

Edit: Also one skintone would be nice

That is mostly down to what you’re painting. I think mostly you should get black, white, primary colors, and then a foundational shade of any color which you will use in large amounts.

A black and a white paint are both very useful. I don’t particularly like the Vallejo white, preferring the Pro-Acryl or Two Thin Coats whites.

Primary colors; red, blue, and yellow are good ideas because while I wouldn’t recommend fully mixing all your own colors, having these colors (along with white) will allow you to mix into most other colors to create highlight shades. I’d recommend bright, saturated shades.

I personally get a lot of use out of VMC Chocolate Brown, Pale grey Blue, and Hull Red. However a lot of that is down to what I’m painting.