Honestly, most of this stuff is going to be covered in any Discrete Math text book. I can recommend: Essentials of Discrete Math, https://www.amazon.com/Essentials-Discrete-Mathematics-Bartlett-Inernational/dp/1449604420 It will also have notes on o-notation, etc.
Yeah, I've heard the same things about Rosen, but I wanted to mention it since the video lectures use Rosen.
The book that the discrete class I'm taking next semester will be using is "Essentials of Discrete Mathematics" by David J. Hunter. From what I have seen, looking through the text, it covers many of the basic materials in a Discrete class. This should be an okay book to use if you find that Rosen isn't to your liking.