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I've found some good colors in the tamiya spray paint range for my tyranid army.
But the question is, do they work as primer?

Or do I have to use a special primer spray (white/black) FIRST,
before I apply the "base color" with the tamiya color spray paint?
I don't want the paint to start falling of my minis....

Paint is paint, primer is primer. Always prime first.

Are they acrylic or enamel?

Always prime with a primer no matter what. Primer is formulated to help the paint bond to the surface of the model.

A dude at the local hobby store handed me a can of yellow tamiya spray that ended up being very glossy and super difficult to paint over. I would look into Army painter because I understand their product has a great matte finish.

It ended up glossy because you most likely laid it on too thick, it'll do that. Very thin coats are necessary, Tamiya is a great brand and provides a great finish when applied properly. One of the smoothest paints on the market, as long as you know how to use it. Every different paint brand acts a little differently, especially when spraying. Tamiya is a much higher quality paint than army painter.