“Since so many history books are tainted by one form of prejudice or another, I have frequently been asked if I know of any more objective works. I have turned for an answer to the only historian on whom I rely completely. Dr. Steve Jolivette, an Objectivist scholar of the rise of the West, and a historical generalist, has looked hard for good books that deal with history broadly—the survey textbooks and certain other books that we all need so badly. This is his short list. These books are not perfect, but they are the best available to us in this era.” (Leonard Peikoff)
For Western Civs textbooks, I can recommend the W.W. Norton publication, Western Civilizations. The latest edition that I have is #12, by Lerner, Meacham, and Burns. It started out in 1941 with Edward McNall Burns being the sole author (they still name him as an author, though he is long deceased and other authors have been added). It has long been popular, and the best parts of it have remained remarkably unaltered (though I would always favor earlier editions, of which there are many inexpensive ones online). In later editions, for politically correct reasons, they have added some stuff about slavery and women. (LP describes Burns as “a deservedly eminent historian.” [DIM Hypothesis, 154])
You might want to try more than one textbook. They are very inexpensive on the online used book lists. (My used book site of choice is www.addall.com; also www.amazon.com may have copies which Addall does not.) I do not mean reading them all cover to cover, but when you find something interesting or something you don’t think is covered well, you might try seeing how it is treated in another book. Another favorite of mine is Stewart Easton, The Heritage of the Past: From Earliest Times to 1500 (I have the 3rd edition). (LP cites Easton several times in DIM.) Another which looks very good, and perhaps the best of all, but which I have not read enough to rank precisely, is Perry, Chase, Jacob, Jacob, and von Laue, Western Civilization: Ideas, Politics & Society (I have the 5th edition).
Consider also W.T. Jones, A History of Western Philosophy (several volumes). It is not a history book, but he has short introductions to each major section of the book, which often are some of the best historical overviews of that period that I know of. The book was reviewed by LP in The Objectivist Newsletter, vol. 3, no. 9 (Sept., 1964): 36.
For a single book on world history, I recommend William H. McNeill, A World History (I have the 3rd edition). It leaves much to be desired, but is the best that I know of (and is renowned), and does cover both the West and the Rest.
For the history of political thought, George Sabine, A History of Political Theory, is clearly the best (it is rare to be able to say “clearly” about any history book recommendation). (Ayn Rand recommended this book. [Scott McConnell, 100 Voices, 154]. LP regards Sabine as “one of the best historians of our time” in DIM, 155.) But Sabine is not for novices. And be careful, because he is off the mark on the issue of individual rights (17th and 18th century).
For the history of philosophy, W.T. Jones (see above) is definitely the best (again, a luxury to be able to say this).
I also recommend poking around the internet for items of interest to you or which you think are not covered well, or which you still have big questions about: Wikipedia (I use it all the time, to good effect), encyclopedias of various kinds, and a host of other sources. Of course, none of these sources (including the books recommended above), are consistently good or even necessarily fully accurate on all points, but many of them are good on the whole.
On this website, I will not praise the obvious for their value not only as philosophy, but also as history. I refer to TOP and DIM.
Best wishes for good reading,