I have an answer that I wrote which goes over the pressure point "knock out" phenomenon:
How do you knock someone out using pressure points?
In summary, the pressure point knock-out stuff is either over-hyped or outright nonsense. There are only a few areas on the body which when hit will produce a knock-out. But all of the complicated TCM (traditional Chinese medicine) / acupuncture / kyusho-jitsu / dim-mak theory is a useless waste of time trying to understand. And getting a KO from these points isn't easy or reliable, as I explained in my answer at the link above.
Of course there are more sensitive places (so called "weak points") on the body that hurt a lot more when you hit there. So you'll target those areas when boxing, kick-boxing, doing karate, etc. There's nothing mystical about it. They're places like the floating ribs, the solar plexus, the groin, the back of the knee, the jaw, the nose, etc. These targets are completely incorporated into most percussive martial arts.
Pressure points are also used in "pain compliance techniques". For example, you can jam your thumb in the hollow behind the ear near the jawbone. It causes pain, which can make someone stop what they're doing and switch to something else. Same with wrist locks, whereby you can press on sensitive areas in the wrist to cause them to release their grip.
Pain compliance techniques are, like percussive martial arts, built into the techniques in grappling based arts. They aren't often the focus of the technique, and so you might not even be told to target these points. There are more important mechanical things you have to be doing correctly first before you can refine it so that you're also getting the pressure points along with it.
I will say this about pain compliance techniques, though: You really can't rely on them one bit. When you're in class with a partner, they work well. But when fighting for real with people that are experiencing adrenaline rushes as a result of the fight-or-flight physiological reaction, it doesn't generally work at all. The reason is that people in this state often don't feel pain. Or it might be very reduced.
Not to mention that in class, people let you do these pain compliance techniques on them, but in real life people don't sit there letting you do stuff to them. They're going to flail around. And while you're trying to crank with one hand on their wrist's pressure points and the other hand on their hand, they have one hand free to pop you in the nose.
The focus, therefore, can't be on those pressure points. What you're doing has to work by itself anyway. The pain compliance stuff comes in only as a refinement if it doesn't jeopardize the original technique in any way. That way, if it works, great. If it doesn't, nothing is lost.
But to answer the question of how much time you should spend in your martial arts practice on studying and targeting pressure points, the answer is zero. That's because your martial art already targets those areas. And so you don't have to spend any extra time isolating that aspect of your training and really drilling it. It should already be a part of it.
Don't go down the wrong path of thinking you're going to go out and learn pressure point striking more seriously so that you'll have some kind of advantage over everyone else in your style. You might be thinking that if you could just pull these techniques out of your tool chest in sparring, you'll be unstoppable. The reality is that if your martial art doesn't already focus on this and doesn't already give you these tools, it's probably because those tools aren't important. Or I should say, they aren't the most important thing you should be learning right now.
The topic of what works for self-defense is much more broad. Targeting weak points on the body is part of the full picture, but a small part, and one of many. Primarily, what works for self-defense is to train in such a way that you are able to test what you know on live, non-compliant partners that are trying all they can to resist you.
It's called "pressure testing", and a good example of this is how MMA people train. If you can make that way of training your foundation, you'll be able to see that targeting pressure points / weak points is more of a refinement of existing methods rather than a supplemental thing you have to go out and learn separately.
And if by some chance you practice a martial art that really doesn't address weak points on the body, they probably have a reason for that. For example, in Judo you don't see a lot of people doing an uchi-mata nage, for example, with the intent of smashing their opponent's groin "by accident" as they're doing it. They're not landing people on the back of their heads to get a knock-out, either. Getting knock-outs and hurting people isn't the goal of Judo. Their goal is to unbalance while standing, to throw, and to control on the ground. If something isn't helping them achieve that goal, they don't spend any time on it.
But Judo does have chokes, they have techniques such as kesa-gatame that restrict breathing, some of their holds can snap the spine with the slightest pressure, etc. Those techniques actually rely on weak points on the body. So even in Judo, these weak areas of the body are incorporated into what they do. But they're not going to spend time isolating them and going over all the ways of utilizing them.
In conclusion, knowing how to utilize pressure points isn't going to give you much. You can study them intellectually, as that might interest you. But there are a lot of other things more important that you need to get first. And even after that, hitting on pressure points will never be your actual goal. You'll still use them, but only if it assists what you're already doing. And when you do use them, you won't even think about it, because it's already incorporated into your technique.
Hope that helps.