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I agree good wash in detergent.
I use a matte primer from Halfords as it’s cheaper and sticks very well to both metal and plastic miniatures.
I personally use a lot of citadel paints as they are easily available in the UK. They’ve seemed to stand up to the kid’s handling them also. They also do a contrast range which may appeal, there are plenty of videos on YouTube about them.
I believe they also do undercoat sprays in certain colours but I believe they are a bit pricey.
I also agree with the gloss followed by matte varnish but I haven’t done that to any of mine as of yet and the paint still holds.
Hope this helps. Happy painting
A small, but important point that I'd suggest in addition to Alan's and Jimmy's suggestion for a good wash is a good rinse afterwards. I triple rinse mine afterwards, in warm and/or cool water, to be sure to remove any residual detergent. I find it crucial prior to undercoating, particularly as I use acrylic paint.
The old trick of mixing a bit of PVA glue with your paint may be a good idea in your case. Just a little bit applied as a wash before the undercoat or with the undercoat. You don't want it to become gluggy!