Do I not even need a book as the internet has all that I need?

Nearly all worthwhile historical scholarship will be in either academic journals or books. Full stop. You need both. If you're a university student, access stuff through your university library - they should have access to most major journals and plenty of ebooks as well as whatever books they have physically. Wikipedia, r/AskHistorians or wherever else isn't a substitute for reading legitimate research and scholarship.

Do I just start reading books about these particular periods of time in history and go on from there?

This is the best place to start. I'd recommend checking out the booklist this sub maintains. There's a lot of accessible and really great works of scholarship about American history and military history there!

If you want more more expansive lists of books and articles to read, try this collection of graduate history reading lists maintained by the University of Wisconsin-Madison.

"Read a bunch of academic books," while dead-on accurate for above average knowledge+specialization, isn't very useful as a starting point--mostly because academic books are purposefully geared for someone waaaay past the starting point. It's tough because you wind up in a place where it really seems like "you have to know something before you can learn it."

My entire academic career has been me falling off a cliff into new areas of study, so, naturally I have plenty ideas of how to do this RIGHT instead of wrong like I do. :D

First--yes, I think it's a good idea to start by reading, say, the Wikipedia article for the broad broad topic. Now, the article is going to be (a) dry, maybe boring, (b) probably long, (c) a mix of intricate details and stuff you want to skim because you already know. The key is--don't get bogged down. Don't take notes; don't try to memorize anything. You aren't trying to learn details right now. You want the General Framework--a timeline, the major players (e.g. which countries are on which sides in WW2, not, like, a full list of colonial American leaders and their biographical details), the most basic broad contours of the situation. You are going to want to know that Germany starts the thing by invading Poland, although Japan is already at war with China; the U.S. gets involved at Pearl Harbor; meanwhile, Germany is slaughtering Europe's Jewish and Romani populations.

So essentially, you just want to acquire the scaffolding or outlines that you are going to slot all the stuff you actually learn into.

If you can find a good pop/academic crossover history on your topic, that's a good place to go next. The AskHistorians booklist will usually say, in the descriptions of the book, what's a good place to start. (Mann's 1491 for the European imperialism era, for example.) This will also probably be a reasonably broad overview, not something to stress about memorizing.

...Actually, "don't stress about memorizing" is THE most important lesson here. Don't bother. It will be stress and make things unfun. Don't worry--as long as you keep reading, you will eventually realize you legit know the stuff (recognize the names, places, events, start to make connections the authors don't) on your own. It's...it's a really cool feeling, actually.

Anyway, if you have a decent enough General Framework, you can start reading more specialized stuff to begin to fill it in. Here's where it can get tricky. The AH booklist is good; you can ask for more direct recommendations on sub-topics that interest you (say you really love reading about women war heroes; maybe you're into home front stories; inspired by one book later on, you develop an interest in scientific progress resulting from military innovations in wartime).

My personal favorite strategy is to see if I can find a PhD exams reading list on a topic. "World War II" might be difficult, since that's not very often a focus of PhD research these days, but "20th century military/political history," perhaps, or "European history" or some such. For U.S. Civil War, you can probably remix a list out of stuff on 19th century U.S. history, African-American History, maybe Southern history...etc.

Many of the books will be from academic publishers and very very expensive. If you're in the U.S., even your public library should have access to Interlibrary Loan, through which you can get a short-term checkout of academic books from university libraries (hopefully). Otherwise, if you live near a major uni, you can most likely use the library in-house--meaning physically be in it--as a guest. Or you might have...your own...ways of obtaining books (say, ebooks in particular) without a university connection.

Anyway, if I were going to start a new area of study systematically, that's how I would go about it.

I hope this is helpful--please let me know if there's anything I need to clarify or if you have further questions. :)

I took a course on a whim as part of my history degree: The History of the Supernatural. Before that, I was mostly interested in the interaction between religion and culture with a History from Below lens - basically, studying history happens through historiography, of which there are a number of schools. The most famous outside of people who do history are probably "Big Man" history, Marxist history, and Feminist history. I was interested in History from Below, which is pretty much the opposite of Big Man history.

Anyway, my lecturer for Hist Supernatural (hi, u/drcmillar!) specialises in early modern English Witchcraft, through the lens of the History of Emotions, and was super interesting, and inspiring. I was still flitting between Marxist historiography and History from Below at this stage, but my focus shifted to religious history in early modern England, and so I started taking more courses in that general direction, rather than the SE Asian stuff I had been doing.

Then I got an opportunity to do a Summer Research program under Dr. Millar, and my slight interest in History of Emotions became a strong interest.

Basically, I took a course that i thought sounded fun and got hooked.

Depends on whether you want to learn as a layman or specialise as a historian. Books written for a popular audience are a great way to learn more as a layman, but if you want to specialise as a member of the field then really there's no substitute for taking a university course and reading as many academic papers on your area of interest as possible.