I’m looking for a book that can serve as a good introduction to Chinese history. Something that lays out different ages and dynasties well, then leads up to the PRC. If there are two separate books that deal with ancient and modern China better on their own that’s okay too.
Well, good luck with that. You're talking about a region the size of Western Europe, over the course of about 2200 years at the least. A single book will not cut it. I'll repeat my standard recommendation of the History of Imperial China series edited by Timothy Brook, a series of six books covering the Qin and Han, the Northern and Southern period, the Sui and Tang, the Song, the Yuan and Ming, and finally the Qing, respectively. After that, a classic standard work is Jonathan Spence's The Search for Modern China, which covers roughly 1600 to 2000.
As you asked for best "overview," here's my personal opinion on the matter: for my (non-Chinese) students, I have always encouraged them to approach Chinese history from the "internal gaze" of classical Chinese historians themselves -- not for the sake of obtaining the most "up-to-date" and "nuanced" understandings of historical moments (as we are talking about introductory overviews here) -- but rather to get a rough sense of how Chinese historiography works. I cannot stress the importance of understanding traditional Chinese historiography as a prerequisite of properly understanding Chinese history from a scholarly point of view. Not only did Chinese historiography took a very distinct development path from the types of historical studies we commonly encounter in modern Western academic institutions, it remains tremendously influential in terms of shaping both present day domestic understandings of Chinese history at both vernacular and scholarly levels.
That being said, when it comes to the actual reading list, the biggest constraint is that good English translations of primary classical Chinese historical texts remain rare and patchy.
To get a good first-impression of Chinese historiography, start with works on pre-imperial (pre-Qin dynasty) periods first. A good place begin would be the Shiji by Sima Qian: you could access Herbert J. Allen's translations of the first three volumes of Shiji for free at Chinese Text Project. The Allen translation is dated but satisfactory in my opinion. He did a good job retaining the accuracy of the terms, although rhetorical style and vocabulary choice have been greatly simplified. As you read through the first three volumes, do not get too bogged down on individual details, but instead try to distill an over-arching ethical "master-narrative" implicit throughout Sima Qian's writing, especially concerning the transformation from quasi-legendary pre-dynastic rulers to the emergence of (also quasi-legendary) Xia dynasty, and on the cycles of rise, mid-crisis, revitalization, and final downfall of Xia and Shang "dynasties."
The second work I'd recommend is Zizhi Tongjian (資治通鑑, or "Comprehensive Mirror in Aid of Governance") by Song historian Sima Guan (1019 - 1086, and whose shared family name with Sima Qian is purely coincidental). Zizhi Tongjian is an exceptionally polished collection that is also absurdly long, and again I would suggest you to begin by only focusing on the first eight volumes of the collection on Warring States and Qin periods. The only English translation of volumes 1-8 of Zizhi Tongjian available is theone by Joseph Yap published in 2016.