Hi, the almighty /u/Jasfss suggested The Great Battles of Antiquity: A Strategic and Tactical Guide to Great Battles That Shaped the Development of War by Gabriel & Boose to me a while ago when I was interested in a related subject. I'll just pass along the message and suggest you pick up this book from a library (it's not really worth its store price). The authors start off the book with an introduction chapter describing the evolution of armor and weapons in the ancient world and then goes on to separate chapters for various battles, which of course is what you're asking about. It's easy to find specific sections in the book, so you can easily read about Greeks, Romans and Chinese military without having to deal with Sumerian, Egyptian, Korean or Japanese battles on top of those. ;)

On the Romans, Adrian Goldsworthy's The Roman Army at War is probably going to be your best account. It is very much in the style of John Keegan.