As the title says I am looking for some good books or other sources that are trustworthy about any part of Roman history. It can be during the Republic, during the Empire or Late Empire or even Byzantine.
For Byzantine history, I think either Judith Herrin's Byzantium: The Surprising Life of a Medieval Empire (2008) or Averil Cameron's The Byzantines (2009) is your best bet. They both write very well for a general audience and they are genuinely some of the biggest names in Byzantine studies, so they are, in my opinion, the two best overviews of Byzantine history. Peter Sarris' Empires of Faith: The Fall of Rome to the Rise of Islam, 500-700 (2011), Mark Whittow's The Making of Byzantium, 600-1025 (1996), and Michael Angold's The Byzantine Empire, 1025-1204: a Political History (1997) are meatier books by specialists and more geared towards students making their first forays into Byzantium, but they are still narrative histories and so should still be relatively accessible if you want more details on specific periods of Byzantine history.
There are recommendations in the Ancient Rome section of the AskHistorians Booklist that might interest you.