Honestly, In learning C++ i never picked up a book (no flaming please).  The best advice i can give is to go to this page and go through the tutorial.  It covers the majority of the C++ language (read: most commonly used features) and keeps it as simple as possible.  As far as APIs that are important... well that is a matter of preference.  No one toolkit/api has really "won", but Qt, GTK-- (gtkmm), and wxWidgets are all big players.  And besides GUIs, you'll probably want to learn either the raw winsock2 and threading APIs OR the boost library's threading and network interfaces.  I do agree that MFC is dying, and for windows-only development C# is taking an increasingly large role (even on linux/mono C# is starting to catch on... slowly).  
Also, the best way to learn a language is to code.  So don't just read a whole ton- without practical experience you're never going to learn the language.  Ask questions, answer those you can, and write tutorials- for yourself if nobody else.  Writing down what you've learned is a great reference, and the process of straightening everything out in your head to write it down in a fashion that another person can understand it alone is a great way to reinforce concepts.  In a strange, but seemingly backwards, way I found the best way I learn programming is by helping other people with their questions.
@Neil- i disagree that online tutorials are "flat out wrong".  If anything, the style they teach you might be geared towards readability rather than optimization- which in-my-not-so-humble-opinion is an advantage.  In my limited experience I've found cplusplus.com a go-to reference for pretty much everything.
Specifically, to answer your questions:
1.  No thick book needed.  I've heard that the thinner ones can be useful as a quick reference, though.
- It's completely subjective and depends on your goal.  Boost's network and multithreading libraries are probably a good start. 
- between std::string, std::stringstream, and getline(std::istream&, std::string&) you should be pretty good.  C++ comes with a whole ton of built-in functionality yet isn't too bloated/huge/impossible to learn.  Take advantage of it.  All of the parsing is built in already.