Your question is actually three parts:
1) How do I make a strong punch?
2) Where can I hit someone to have the best result?
3) How do I deal with multiple attackers?
My answer to the first question is: Find a boxing gym and have them teach you how to punch hard without breaking your hand. Wrist strength is what you seem to think is the most important thing, but it's just one link in a long chain of things you need to learn in order to make a strong punch.
For starters, you need to know how to make a proper fist. An improperly formed fist will result in broken fingers if you punch something hard. Second, you must learn how to stand properly. If you throw the punch with a stance that's not balanced, you can lose your balance and fall. Third, you have to learn how to shift your body weight so that you can utilize your whole body in the punch. Punches that come from the whole body are much more powerful than those that use only the arm muscles. Fourth, you need to practice punching on a punching bag. Punching to the air won't prepare you for actually hitting something hard. The punching bag will give you feedback on everything. You will find out the limits of how hard you can punch while not damaging your hand. And it well test your balance and how good your stance is (many people actually wobble around the first time they punch a punching bag). Fifth, it doesn't matter how strong your punch is if you can't hit the target. So you must learn how to target a moving opponent that doesn't want to be hit. Sixth, there are many ways to punch (jab, hook, straight, cross, upper cut, corkscrew, hammer, back-fist, etc.). Each punch can hit hard. And each punch is for a particular situation. Learning just one of those punches isn't typically enough, because your opponent will move around, making your window of opportunity to use that particular punch small. Seventh, you'll need to learn how to defend against punches and take-downs. Nobody will stand there and let you hit them. They'll fight back. The best punch in the world does you no good if you get knocked down before you can land it.
I could go on. But the point is that you need to get to a boxing gym to learn what they have to teach you. Just learning one type of punch and learning it without an instructor and without partners, you will have learned almost nothing.
My answer to the second question is: Anywhere you can land a punch is a good place to hit. That's it. Very simple.
The trouble is, people move around. They flinch when they see something coming close to their heads. They cover up and block. So if you see a clear opening, you take it. It doesn't matter if it's not the best place on the body to hit.
That being said, there are some weak spots on the body: jaw, temple, nose, back of the head at the base of the skull, solar plexus, floating ribs, kidneys, and groin. You can find these "vital points" listed in just about all martial arts books. There's nothing mystical about them. They represent places on the body which aren't protected by solid bone. They're weak areas. Hit them, and you'll hurt your opponent more.
For punches, your best bet is the nose or jaw line. Following that, the solar plexus. Those end fights.
But once again, just knowing this does you very little good. You first need to know how to target a moving opponent. They're not going to stand there letting you hit them. And they will hit back. So get to a boxing gym to learn how it's done.
My answer to your third question is: Multiple attackers? Most likely you're asking because you want to defend yourself from someone else who is bullying you and has friends who come with him? If that is the case, you need to realize that this situation is nearly impossible to win. If you start fighting with one of them, the other two can easily sucker punch you when your back is turned to them. It's inviting disaster. This is not going to work out well for you.
So don't allow that situation to happen. Multiple attackers isn't something even advanced level martial artists are very likely to win against. Some martial arts have strategies for dealing with multiple attackers. I trained in them myself, and I now see them as mostly wishful thinking. They might improve your odds by some small amount. Better than nothing I guess.
That being said, the primary strategy for dealing with multiple attackers is to run away the first chance you can get. If you can't run, move to a position where one person is closer than the other two, while still being able to move around. That's the guy you fight. You turn it into a one-on-one fight instead of having to deal with 3 people at the same time.
But it won't take long before the other two move around and get to you. So you have to do this continuously. You need to see everyone and be aware of when you're about to be in danger. Anticipate it and move so that you can continue to engage with only one person. If that means you have to change to fighting another guy instead of the first guy, then that's what you do. While you're doing this, move towards a more open area and with more ways to escape. And the moment you see a way out, run.
Like I said, it's mostly wishful thinking. No martial art deals well with the multiple attacker scenario. None of them.
It sounds like you think you can just quickly punch at their weak points in a preemptive attack to knock them all down before they realize what's going on. That won't work. You're lucky if you can stun the first guy. He'll recover in a second or two and will be very angry. Meanwhile, the other two guys will now be ready for you, and they might hold you until the first guy recovers. It's just not realistic to believe you can perfectly target and knock out all three guys in less than a second. You would have to be Superman or Flash.
Lastly, I'd just like to say that if you're being bullied, you should seriously look into the Gracie Jiujitsu "Bullyproof Program". It's for kids ages 5 through 13. But adults who are being bullied should also check out the web site:
If you can find Gracie Jiujitsu schools in your area, you might want to visit them also. In addition to looking into boxing classes.
Hope that helps.