1) What is the best Linux IDE (In Your Opinion)
The one you are most productive with (ranging from terminal text editor + compiler to full-blown MSVC inside wine, though the latter might be kind of self-defeating). Personally I use C::B for my C/C++ projects mainly because it just flows more naturally for me than text editor -> makefile, though this might change in the future. YMMV.
2) What are the Linux 101 programs I need to get right-away?
Most of the baseline stuff is normally prepackaged in the distribution, and any extras you need are highly subjective. For instance I always grab Skype and Code::Blocks when I install a new distribution, among others. What do you need? Mint has a software manager GUI which you can use to quickly get any big applications you need (like IDE's, etc...) without needing to look up their exact names through apt-get. I'm not sure if g++ is shipped with Mint, I recall not having to install it but then it might've come along with C::B...
3) Is using the shell good for programming
Takes a while to get used to it, it's useful to automate, do various tasks, schedule stuff, generally perform maintenance, etc... it's not at all like Windows and the shell is actually useful in Linux (yes, yes, it can come in handy in Windows too but not at all at the same level). I'd say definitely learn it (but don't waste all your time on it either - shell is not everything, contrary to what some linux gurus would have you believe, there is just some stuff that is simply better served by graphical interfaces - but it is important)