The History of Rome podcast by Mike Duncan is quite good for a broad overview of Rome from its founding to right after Constantine. He uses many reliable sources and often discusses the various theories and ideas and where they come from. Very good for someone looking for a general introduction.

People might recommend a book, I would recommend starting with Wikipedia. Sometimes it's easiest to learn about history by creating a broad framework for knowledge. Learn the basic timeframe for Roman civilization- kingdom, republic (early, middle [ie, Punic Wars], late [ie, civil wars]) and empire. If you don't like memorizing dates, that's ok, just become acquainted with the basic narrative of each era.

Once you've done that, I would pick up The Romans: From Village to Empire. It's a great book in a textbook format. It'll fill you in on the course of Roman antiquity.

Additionally, if you want to follow my suggestions, I would pause learning about any further Roman history when you get to Constantine. Late Antiquity is exceptionally messy and I would only pursue that once you feel pretty comfortable with the rest of Roman history (though be sure to return to it!). Also, don't neglect the Byzantines, but again that's another thousand-year stretch I'd save for later. Good luck!

http://www.reddit.com/r/AskHistorians/wiki/books/europe#wiki_ancient_rome

We have a very extensive book list for Roman history, check it out!

'The Roman Republic: A Very Short Introduction' by Dr David Gwynn. I had this guy at university and he's incredible. Also 'The Roman Republic' by Michael Crawford. These books are a good place to start for general overviews of Republican Rome. There is one for the Empire too.

I really enjoy Adrian Goldsworthy. His books are a little more focused, but very accessible.