I'd recommend Enamel Paints, they stick to pretty much anything and have a pretty good glossy coat. Major downside is it the cheaper brands tend to leave crazy brushstrokes. I occasionally mix mine with acrylics to make that less of an issue, though I'm not much of an expert. If you still have trouble depending on what the case is made out of you may want to sand it down a tiny bit with a very low grain sandpaper that'll make it more cooperative with any paints you use.
In my limited experience it's good to buy an extra cheap one to figure out what methods work on that before moving to a higher quality case.
I'm assuming it's a hard plastic case? If so, it shouldn't be too much different than painting a plastic model:
- wash it in soapy water (removes manufacturer's mold release and any other oils) - sand lightly with a fine grit sandpaper (Optional, but the stock case will probably have a super-smooth surface, this will help paint/primer stick) - spray with a good primer made for plastic (will make painting 100% easier as opposed to bare plastic - trust me on this) - paint with model-grade acrylics (I'd use either vallejo or reaper master series - easier to work with than oil-based paints) - lastly, seal with a good clear coat so all your hard work doesn't rub off. Up to this point I've listed pretty standard modeling supplies you can find in most hobby stores, but I'd go to the hardware store to get a good spray-on clearcoat. You'll need something a little more heavy-duty than what you'd use on a typical plastic model. After all, the phone's going to be carried around in someone's pocket/bag all the time...
As far as silicone/rubber flexible case go... I wouldn't even know where to start, sorry. Seems like any medium you would apply would probably just flake/rub off the first time the thing flexed. I dunno though, maybe someone else knows how to do this...
Anyway, good luck, and be sure to post the results, would love to see it!
You can use acrylic hobby paint (water base) since it's easier to work with than oil based ones but afterwards to protect it from chipping a coat of varnish (matt type if you don't want a shine) will do wonders. If you are going to use a plastic case be sure to give it a bath with a lot of standard washing up liquid, that takes away the fatty layer that most plastic contains on the surface.
If you have a hobby store nearby I am pretty sure they can help you out