So for general surveys:

With qualifications I'd recommend Michael J. Seth's (500-pages-concise) A Concise History of Korea as the best place to start. It's relatively expensive but there's also a volume just covering the premodern part, or alternatively he also wrote the much cheaper and actually concise Korea: A Very Short Introduction which I've only glanced through but looks like a much briefer version of the same material. In general Seth does a good job keeping things approachable for unfamiliar audiences and, vitally, covers Korean history's abundance of controversies about as evenhandedly as is possible to do. There are a fair few things here and there I take issue with (he uses some questionable demographic figures, and affords Mimana more credit than I see most scholars give it these days, for instance) but much less than any of the other general surveys out there, and it's also extremely up to date.

That said, while I think what Seth has on it is serviceable, ancient stuff (pre-Goryeo in particular) is clearly not his specialty in the same way. So for that I'd recommend Gina Barnes' The Rise of Civilization in China, Korea, and Japan (she has 3 editions; make sure you look for the 2015 one). It's obviously not just focused on Korea, but for the ancient period especially a cross regional perspective is incredibly helpful -- and, incidentally, I wish I could give that same advice to people interested in approaching Chinese or Japanese history too; things get way too insular in East Asian studies. Fair warning though, this book is heavy on archaeology, but I think what it provides is vital and unfortunately unique at the present time.

The third pick and the old standard is the classic A New History of Korea by Yi Ki-baek, but its newest edition is almost 40 years old now so the field has moved on enormously; Korean archaeology was basically not a thing when it was written, and incorporation of that in more recent times has almost revolutionized how ancient Korean history is understood. Where Yi's work is still strong though is its character focus; Seth often only glances on Korea's major figures for a moment, whereas Yi will go into some depth, so if you want to know the big names in Korean history it offers something. It is also pretty easy to get a hold of for cheap these days.

Final bonus pick: after a 25 year wait the 4 volumes of The Cambridge History of Korea are finally supposed to be coming out this autumn, so if you're willing to wait I would probably advise going for them, assuming it's anything like the quality of the other Cambridge Histories. I think all three books I listed have their merits, but the sad reality is that there isn't really any definitive book on Korean history out there at the moment, and the release of these will hopefully change that.

For more specific topics there are English language books that are out now that I'd recommend much more whole heartedly, if there's a topic you have in mind. Women in Korean History, covers women's life and status and a bunch of important and under-appreciated figures; Everyday Life in Joseon-Era Korea has fascinating essays on a wide variety of parts of Joseon life of all social classes; Korea's classic histories The Samguk Yusa and 3 separate volumes of the Samguk Sagi all have very good translations for a more narrative history of Korea's ancient past; Empire's Twilight by David Robinson presents a fascinating cross-regional perspective of Goryeo in the dying Mongol Empire, touching the little known period of time when Korea had a profound cultural influence on China.