I'd say C is probably the be-all end-all for Linux. I mean, C was created for the purpose of writing the original implementation of Unix back in the '60's, so C is VERY MUCH ingrained in the *nix scene. Perl is also a biggie as it lets you rapidly manipulate large sets of data. As I understand it, Perl is pretty much the go-to language for any kind of complex string manipulation. PHP might be a good thing to pick up for Linux if you want to do any web stuff as it is part of the heart and soul of LAMP (Linux, Apache, MySQL, and PHP).
If you are interested in Kernel development or OS development in general, then C is an absolute MUST and the assembly language of your target processor should be fairly useful as well.
From what I heard, programming in Python can be pretty fun as you can use it to make prototypes for "real programs" (read: C-programs) really fast. There are decent tools for Python as I understand it (though I haven't used any) and as for its relation to Linux, Gentoo uses it to develop their package manager, so I think that means it carries some importance in the Linux world.
If you want a really handy tool, learn to program in the Shell (various dialects such as bash, ksh, zsh). It is really useful for making tools to automate tasks, and you can then use a cron program to automatically run those tools at a certain time!
If you would like to contribute to the KDE project, your C++ will serve you well. This is just something to keep in mind. It should also be fairly easy to make C++ programs for Linux, unless of course you learned to do C++ using something like visual studio or borland where much of the code is generated for you.
Java: this is a bit of a weird one as it has been in various stages of becoming open source for a while. It is good for making cross platform programs as it is an "interpreted" language (compiled to bytecode which is later interpreted at run time by a virtual machine). Programming in Java will be a bit disorienting if you are a big C/C++ fan as they remove pointers! Java was actually my first "real language" (the BASIC I knew doesn't really count), and though I liked it, after years of C/C++, going back to Java felt a bit like being given a pair of blunt scissors to work with.
As for development environments, if it is an IDE you want Eclipse is a good bet as it has only about 10^9 - 1 plugins for it. If you want to be a little more "hardcore" you can just choose to invoke gcc manually and program in vim and learn to use gdb from the command line (or use ddd, which is a frontend).
For a really awesome editing tool, you should really learn to use vim. I can't emphasize this enough! Once I started using vim, I was able to finish long and tedious editing tasks in a fraction of the time. Case in point being trying to fix code that got mangled by my programming partner's editor. His editor screwed up all the tabs and made our code hard to read. Normally I'd have to go through and fix everything manually, but in vim, all I did was type about 10 keys to combine different commands and it fixed the formatting of the entire file!
Vim is a great editor, and if you'd rather try a graphical version of it, you can install gVim instead, which also has Windows and Mac support. It comes with a bunch of pulldown menus at the top and all the commands in the menus have their keyboard shortcuts listed for a quick reference. Vim is intimidating to learn at first, but after about a week or so of just using it, you will probably never want to go back to notepad or gedit or whatever. If, on the other hand, you don't like vim, I have heard good things about nano/pico, joe, emacs, and gedit is pretty useful as it has tabs and syntax highlighting (like vim).
There are SO MANY free tools for programming on Linux that you are sure to find SOMETHING that you will like. Just try out the different tools we suggested and go nuts!
Just to recap in case you got bored and skipped to the end: C is the heart and soul of Unix, vim is a great editor, Perl/Python/PHP are all good languages to pick up, and Java is weird. I hope this helps!
-Zorak