Well, this can probably go for all Fallout-games, but since I have really only played F3, I focus on that in this post.

In the DC Wasteland, the people are few and most of them are common people just trying to get by from day to day. But in bigger settlements like Megaton and Rivet City you get specialised people like guards, scientists and doctors. Wouldn't it be vital for the survival and development of the human race that those that had specialized knowledge in any field useful for survival would teach that to others as well? Especially scientists like Dr. Li and the doctors should divide their time between doing their daily job and having teaching classes. Not only them, but also those that are able to use weapons, tame and care for Brahmin and build livable houses and shelter.

Is there any in-game lore about this? What do you guys think?

Ther is a shcool in the rebuplic of dave but they arnt getting books any more and I'd immagine most people do like lucas simms and teach their children themselves so dr.li's mother was probaly a doctor and dr.li learned from her.

so children dont realy get a choice as to what they want to do, they just learn from what their arournd

It sounds a bit hazardous, though, to only teach your children. There is a big chance for imporant knowledge getting lost then.

Thats probaly why there are so may raiders, beggers, and wanderars. becuase they never learned other things to do with their life.

Well... after the purifier comes up Rivet city and the surrounding area will become a trade hub. Trade hubs with access to water, and protection usually become civilized... sooner or later. So I would expect a rise in people migrating in to be close to the fresh water and then children there would be educated in some way shape or form.

There was life and civilisation before universal education, in fact before education of any kind.

After the bomb, Priorities 1,2 and 3 would be immediate survival. Educating the young doesnt fit in any of those. You'd be looking at immediate medical care, Getting immediate food supplies, Immediate access to clean water and shelter.

After the immediate needs are met, then some sort of education can begin. I'd imagine in the DC wasteland, and in other "Non-Civilised" areas for skilled trades you'd be looking at an unofficial Master-Apprentice system (unoffical in that there's no certificate or set ciricculum, when the master says you're ready to go alone, you go alone.... Or if the apprentice is convinced he has nothing more to learn he goes alone).

In more civilised areas (Vault City, NCR, Rivet city) public education becomes more viable (and in VC's case, I'd Imagine Manditory given its beurocratic nature). Larger towns and communities could have a schoolhouse for children, probably 1-2 teachers for the entire school. Given Rivet City's scientific base and numerous "Role Models" for the children, I'd expect a relatively excellent primary/secondary education system

In VC, I'd imagine Tertairy education for skills trades would probably again be based around a Master-Apprentice system with a regulated/set cirriculum and testing, with hands on experience supplemented with theoretical learning and testing from the Vault Library and education computers.

In San Fran, we know the Doctor there provides training for would-be doctors. (IIRC the Broken Hills doctor was trained by him)

In NCR, I think given NCR's "Statehood/Nationhood" impulses, I think a "University of NCR" would be appropriate. In any case, it would likely be very new, with most "Educated" people starting from the apprentice system, and the University probably more focused on "Practical" trades (Legal, Medical, Engineering, etc) and probably little, if any "pure acedemic" and humanities (History/Anthropology) studies.

Yeah, I wasn't necessarily talking about proper education in an organized school system. But, for instance, I don't think Dr. Li and the doctors in Megaton and Rivet City are fighting every day for food. They do have their primary needs met - food, water, safety and a roof over their heads. And since they have specialized knowledge in subjects that might be vital for people's abilit to evolve, in a situation that is far more precarious than humanity pre-school days, I would have thought teaching their skills to others would be a priority.

Was just general thinking about Education in general, so not all of it applies to your question (such as immediate concerns, tertary education, etc).