Yes you should. It doesn't have to be more than a cover coat, and if you can avoid getting it on the gluing surfaces it makes using solvent type cement easier. You can't bond painted parts with it.
But you can remove paint on gluing surfaces with a quick scrape of a knife, or a swipe with a sanding stick.
Or use superglue. It works on painted parts pretty well.
My favorite primer for models that matter a lot to me is Tamiya white lacquer primer in the rattle can. it's a little spendy though, at $ 9 a can. They sell a less expensive, and not quite as smooth, gray primer too.
It used to be a thing that plastic needed to be washed to remove oils. I don't run into that with models from the big companies like Revell anymore, so I don't bother. Others do.
I like white primer. it's because most colored paints are formulated as a mix of pigments against white. So you get the best true color.
I suppose that white primer in a couple of coats c ould also become the white ship color. No reason why not.
I plan to prime my corvette with Rustoleum Red. Because I plan to 'beat up' the colors to expose the primer all over.