1. Currently using Kate right now, it's a test editor with a lot of features.

2. You need to get g++ or clang, by default mint doesn't come with a c++ compiler.

3. shell is useful in unix-like OS's. The main reason I am using Kate editor is because it has an integrated terminal emulator, so I can edit code on one half of the screen, and compile with makefile and test the program on the other half of the screen without needing to switch windows.

I am using linux mint 13 now ;o

1) I also use and love Code::Blocks. Although I personally don't like it, you might want to give eclipse a try.

2) Linux has so many usefull tools, I guess I should split them up into categories. So these are the tools I use the most for programming related tasks, which doesn't mean you need to get them "right away". But they might prove useful.
FILES:
- SVN or GIT (version control)
- Meld (nice tool for merging files, probably not the best but I like it)
ART:
- Blender (3D Modeling tool)
- Gimp (painting tool)
- Inkscape (drawing tool, also good for editing PDFs)
- mogrify (commandline tool for batch processing of images, for example resizing)
NETWORK:
- Firefox + Google (obviously ^^)
- ssh (ssh client to log into a remote machine)
- netcat nc (usefull tool to send/recieve TCP or UDP messages for debugging purposes)
- Wireshark (very powerful network sniffer)
- Filezilla (nice ftp client)
MATH:
- octave (Matlab clone, a lot slower but usually sufficient)
- wxMaxima (nice tool for solving equations, computing integrals, etc.)
PROGRAMMING:
- Code::Blocks + gcc [+Cuda]
- Valgrind (bunch of tools for profiling and memory leak detection)
- oprof (profiler which uses the CPU's hardware counters)
- kcachegrind (my favorite tool to display profiling information, not sure if it is the best)
- perf top (builds on top of oprof and gives you an overview, which functions in your entire system use the most CPU time right now)
- iotop (tells you, which processes are using your HDD right now)

3) As was states above, the shell in linux is in general more useful then in windows and you should use it. However for programming I recommend an IDE like Code::Blocks or Eclipse.

QtCreator is pretty good and works very well even for non Qt based projects. I use cmake and qt creator handles cmake projects pretty well (with a caveat that can easily be worked around).

It is good because it's fast, with a minimalistic yet useful user interface entirely based on spliting the window into non-overlapping areas, and using drop down lists instead of tabs (which makes it much less messy to juggle between lots of files than in say, visual studio). I used to use kdevelop 4 (which also have a strong support for cmake projects) but it was a bit slow and unstable (I pretty much always had to disable the indexer).

As for other software, if you're doing C/C++ development I can't recommend cmake enough.

You can easily automatize finding out where third party libraries that you need are installed and automatically setup the necessary include and linking paths, you can easily setup unit tests and execute them automatically, you can easily do non trivial build stuff like compiling a custom code generation tool and then run it before compiling its output all with the right dependency order, and last but not least it's cross platform so if some day you want to make a windows version of your project you can still use cmake to do it.

I do the same as Goran. I use gvim and cmake when working in Linux. I have some custom bash aliases setup for making greps easier, but other than that I use a linux console the vast majority of the time. I use ddd or kgdb for debugging. Valgrind (as was mentioned) is great for leak detection.

If I were going to use an IDE though in Linux I would probably use QtCreator or Code::Blocks.

I like Mecurial for source control. Wireshark is an excellent program for debugging network related stuff, but tcpdump can be useful also.

Just don't underestimate the power of a console... the ability to define aliases can be very powerful. Especially when you learn redirection operators. I also write tons of bash scripts to automate things.