I can recommend A History of World Societies which is a work carried out by a large group of historians that act as a great introduction to World History.

With plenty of references and working questions for you to reflect over, it works well at guiding you further along if any particular piece of history interests you. Also it deals with some of the eurocentrist writings of more "classical" world history books.

Does it break completely from eurocentrism? Not at all, but it at least acknowledges its role in acting purely in an introductory manner. Quoting from the preface on p. v of the 11th Edition (2018)

... we have made every effort to strike an effective global and regional balance. The whole world interacts today, and to understand the interactions and what they mean for today's citizens, we must study the whole world's history. Thus we have adopted a comprehensive regional organization with a global perspective that is clear and manageable for students. For example, Chapter 7 introduces students in depth to East Asia, and at the same time the chapter highlights the cultural connections that occured via the Silk Road and the spread of Buddhism. We study all geographical areas, conscious of the separate histories of many parts of the world, particularly in the earlies millenia of human development. We also stress the links among cultures, political units, and economic systems, for these connections have made the world what it is today

It's "only" 1000 pages (with annoyingly thin pages that easily rip..) but should serve as a good introduction to some of the larger events and the role they have played in the world (you mentioning the French Revolution, American Civil War and World War 2) whilst also connecting them to the rest of the world. It's with good reason its used as a teaching book, at least at Copenhagen University.

Are you going to become an expert in every field of history? No that takes dedicating yourself to reading a lot of books. But at least with this one you can decide how you want to proceed with your journey into the wonderful world of history.

In my experience it is really easy to burn out when reading history surveys (books that cover a very large timespan or geography), since the details can be overwhelming and the narrative can be unclear. So, it might be a good idea to switch between topics you already know a bit about and reading surveys, so that even if you find the survey boring, you can still gain some historical knowledge.

For finding sources, I would recommend:

1. Since you are at university, you could check to see what textbooks the university recommends for intro world history.
2. If you don't think your home department is that good, you can also Google to see if there are other universities that have their reading lists or course guides online.
   1. I like using Cambridge, because it puts a lot online. Although if your interest is world history, they only have two courses up: 1400-1914 and 1914-Present.
3. A third option might be to check out H-Net. I know H-German better than the others; it had a syllabus contest in 2015 and has a database of previous syllabi.

I focus on modern German history, but I think a good European history textbook might be The Making of the West by Lynn Hunt, Bonnie Smith, Barbara Rosenwein, and Thomas Martin. The volumes cover the classical to the modern era and each chapter has a list of suggested readings if you want to go further. I found it to be a pretty lively read and the analytic and narrative points to be pretty transparent.

If you're interested in the Cold War and World War II, I would suggest

1. Dark Continent by Mazower. It is a critical read of European history that spans the end of World War I to the end of the Cold War.
2. Postwar by Tony Judt, perhaps as a supplement, which focuses on Europe's recovery from WWII, decolonisation (from the European perspective), the fall of traditional social democracy, and the rise of the Cold War.

I don't know what a particularly good book for the French Revolution or the American Civil War would be. I enjoyed Kates' ed. The French Revolution: Recent Debates, but I think it assumes knowing too much about the Revolution to be a good introduction. I also thought Masur's The Civil War: A Concise History was really good at introducing you to the debates historians have about the Civil War.

I'm not sure I'm allowed to answer as my response isn't academic but fundamentally the scope of your request is too large, there's simply too much history. I'm British and I have a non-academic level of knowledge, I know a lot about the last 2000 years of Britain, a fair bit of European and North American history and 20th century major world events but that means I know next to nothing of African, Asian and South American history. Not only that but there are massive gaps in my knowledge of the subjects that I feel I can talk about casually (for example my knowledge of British Monarchs is a highlight real at best).

Your first port of call should be to identify the areas of history you want to focus on, I suggest you start close to home. Once you know where you want to start I would recommend Wikipedia, it has broad level information on literally everything and will take you on a path of never ending links to things you find interesting. Once you have that broad level of knowledge you can do deep dives into specific periods or events. A basic hierarchy of information would be youtube for bite size knowledge, podcasts for in depth discussions by people who know what they're talking about and books to really get into the weeds of a subject.