Reddit mentions of A Brief History of Chinese and Japanese Civilizations

Best political histories are the Cambridge History of China series. Pretty expensive books, but it's got the best information available in English.

If you're just starting out, check out the wikipedia pages on the various dynasties. That's a good starting point to get you acclimated.

For religious study, try starting with Three Ways of Thought in Ancient China by Arthur Waley. Also check out Sources of Chinese Tradition.

I don't recommend starting with old transmitted texts as Psyqlone suggests because you won't understand the context in which these works were written. Good "survey" books are the best way to start out, then go deeper from there.

Silk Road: Monks, Warriors & Merchants is a great book about the Silk Road and its effect on Central and East Asia. It's an English translation of a French book and it's fantastic.

The Golden Peaches of Samarkand is my favorite. It talks about the various goods imported into China during the Tang dynasty. Specialized book, but just throwing it out there. Is there any particular aspect of Chinese or other Asian culture you are interested in?

As far as a good overview book cover it all, I don't have a lot to recommend to you. In college I had to read A Brief History of Chinese and Japanese Civilizations. At the time, I didn't really read much of it, but I definitely don't remember being blown away by it. The reviews aren't great. On the other hand, you can get it used for only a penny.