Shaving your most sensible parts such as the groin can be done with a regular razor though it's a bit tricky and you can easily cut yourself. I've done it in the past a few times but the skin down there is very sensitive so you will most likely get rashes that will swell and itch a lot. For this reason I switched to using hair removal cream for the body and I find it a much smoother experience. If you decide to use cream though, pay attention to which kind of cream you have. There are creams that must be left for 10 minutes or even more on the zone in question for it to fully take effect and there are creams for which 5 minutes is more than enough. If you leave it on too much you will be in a lot of pain, and you might even get chemical burns which can hurt a lot. When using cream there is a fine line between not leaving it on too much and not getting all the hair removed or leaving it on too much and being in a lot of pain so be very carefull with it and read the instructions carefully.
As for the face, I would never apply cream there because it can go wrong quickly. Just immagine that you move your hand a bit and don't notice it and some of the cream gets on to your head, you will have patches of hair that will fall off. I use a normal razor for the face and have no issues with it whatsoever, just get a good foam and you will be okay. Going against hair-growth is the best way of removing even the stubble. If you get rashes after shaving then it's time to look for a new foam, we all have different skin type so what is good for one might not be good for the other, you just have to try untill you find the perfect one for you.
“Get busy living or get busy dying.”
Stephen King