In addition to reading Stroustrup's books (suggested by another answer here), I'd suggest his two FAQs as a great starting point:
They both link to further reading material.
These two, along with the C++ FAQ Lite, are required reading for new programmers at my workplace.
Once you're even a bit more comfortable, consider jumping in to the community: subscribe to something like the boost mailinglists, watch the blogs of big name figures like Herb Sutter, read Alexandrescu's Guru of the Week articles. You may feel like you're in over your head (I still do often after many years of reading the lists) but you'll learn a ton, especially watching the language grow and evolve. (And this stuff is free!)
You can get more understanding from watching how a language changes over time, and how people actually use it, than you can from a million hours of memorizing the standard. (Not that shelling out a few bucks to have a copy of the ISO/IEC standard around is a bad idea, mind -- great for reference from time to time.)
As for books:
- after the Stroustrup books,
- start with the Meyers ("Effective") books.
- I'd also recommend C++ Coding Standards -- a great amount of work went into gathering the "common sense" of the users of the language in this book.
- Then if you want to blow your mind you can look at Modern C++ Design or some of the fun template metaprogramming resources...
Above all, just stay connected and interested. Mailinglists, blogs, websites, academic papers, magazines, whatever -- choose what suits you best, don't expect to follow everything all the time, but keep your ears and eyes open; find aspects that interest you and follow them!