If you want an IDE, I think most people use
Anjuta. The text editors vi and emacs can also be used as IDEs I've heard.
Without using an IDE, you can write code in a text editor and then save it as filename.cpp (where "filename" is any name you choose). Then, open up a terminal window and cd to the directory in which your file is stored. Typing "g++ filename.cpp" without the quotes will compile the source code and the executable will be called a.out. g++ is the GNU C++ compiler and you need to have the GCC packages installed on your system to be able to use it. To run it, type "./a.out", again without the quotes. If you want the executable to be called something else, such as simply "filename", you need to type "g++ filename.cpp -o filename". Read the man page for g++ by typing "man g++" at the $ or # prompts.
As far as learning the language itself: books and tutorials are the way to go I think. Google for some tutorials. For books, I'm sure others can recommend some too but here are 3:
Thinking in C++, Bruce Eckel - the link points to an online version of both volumes of Eckel's book. I'm not sure how good they are.
Sams Teach Yourself C++ for Linux in 21 Days, Jesse Liberty and David Horvath - I have this and found it a bit hard to follow.
ANSI C++: The Complete Language, Ivor Horton - again I have this book. It's not specifically for Linux, but I find that Horton explains things really well. It's a good book IMO.