Supposedly chalk paint sticks to everything but if you care about the piece I wouldn't take that for granted.
Does the furniture have that smooth melamine or laminate-type finish? If so, I would prime. (But you could test some chalk paint on the back where it won't show, if you really want to skip priming.)
I just did a little Ikea-type cabinet, first priming and then doing an olive green chalk paint. It came out very nice.
I have made some mistakes with chalk paint; here are tips that help me:
a) they say you don't have to prime but it does help
b) the paint is very thick. I was getting bad results until someone instructed me to have a dish of water handy and dip the brush in it frequently, then dip the brush in the paint, to thin it down. You also can thin the paint itself with water but I never bothered. The wet-brush technique is fine. Better to do two thin coats than one thick coat. It dries VERY quickly so don't dawdle.
c) When the paint is fully dry, you can sand it with a very fine grit to remove some of the texture and get a smooth feel, if you like the distressed look you can hit the edges and raised areas harder. I'd wait a day after painting to sand or distress, maybe more if the weather is very wet.
d) you need to seal it so that it doesn't scratch. the chalk paint companies sell waxes and sealers that can be sponged on (one or the other) or you can use a matte clear spray; don't necessarily need to buy their own brand sealer
The chippy look is more what one gets from milk paint, rather than chalk paint. Milk paint is sold as a powder and you mix it to preferrred consistency with water. I have tried it on little pieces of furniture from boot sales and it's a giant hassle.
I quite enjoy these little projects and am improving with each one. You might want to experiment with something before you do your furniture. A tray, or a junky little table, or wooden crate or whatnot. Just to get the brushing technique down.