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General Introduction
Before we start, let me emphasize that if you want to turn this into the ultimate experience, you have to engage with the way IDEA tries to help you. This means that you have to refer to the shortcut keymap and this guide for some days until you are comfortable with the main features. Everything in IDEA is organized in a way that you, generally speaking, won't need your mouse from the moment you have opened your IDE. Therefore, the first thing I want you to do is to click on Help in the menu bar and print the Default Keymap Reference. The short-cuts are similar on all systems, although on Mac OS X there are a few differences. The biggest one is that the Cmd key is used when the other systems use Ctrl. For the guide here, I will use the shortcuts for Linux and Windows which use Ctrl.
Autocompletion is probably the very first feature you will notice when you start to use IDEA. It is implemented in a way that it works without the need for a special shortcut. It will silently pop-up during typing and suggest the most likely function names or variables.
You have several ways to use one of the suggestions from the autocompletion window. If you are happy with the very first one that is selected, then just press Enter or better, use Ctrl+Shift+Enter. The latter is called Smart Enter and does a lot of useful things in different situations. In the case of autocompleting a function name like Plot, it will insert braces for you and place the cursor inside. So when you type P3 and you are happy with the ParametricPlot3D at the top of the suggestion list, then press Ctrl+Shift+Enter and you are left with ParametricPlot3D[] and the cursor between the braces. If the word you are completing is not a function, like PlotRange, which is an option, Smart Enter will just insert the word without braces.
Note that you can use the up- and down-arrows to start navigating in the list of completion suggestions. Usually, this is not necessary because with Camel Humps you can often specify the word uniquely. By the way, camel humps are the upper cases letters in a long function name: For instance PP3D for ParametricPlot3D or LLICP for ListLineIntegralConvolutionPlot. In fact, you don't have to specify every camel hump. You can leave some out and IDEA will happily give you correct suggestions anyway.
You can always invoke autocompletion manually (if you have e.g. pressed Esc to close the autocompletion popup) by using Ctrl+Space. If the completion window is already open, then you will see that the first entry will be highlighted: the completion is selected just like when you use the arrow keys. On such a selected entry in the completion list, you can always use open the documentation of the function using Ctrl+Q. Maybe this is important so let's make a short side-tour right here:
With Ctrl+Q you can open the documentation of the function your cursor is currently over. This works for operators too! Try to write fun::myMsg = "Hello you" and use the shortcut when you are next to the = or between ::. The plugin comes with the documentation of all built-in functions and operators.
Back to how to use autocompletion: We have seen already that you can use Enter to simply insert the completion and Smart Enter (Ctrl+Shift+Enter) to do something more clever as it e.g. inserts braces for you. There is a third option: the Tab key. This is especially useful if you have a word right to your cursor and you want to replace it with the current selection. Let's say you have written GridLinesStyle but you made a mistake because you wanted to have GridBoxSpacings. Do the following, go directly between the letters d and L in GridLinesStyle and type BS. When the correct suggestion pops up, press Tab and see that the rest of the word is replaced.
There are two more things which are extremely helpful. The first is called Smart Completion and works only in specific places. You can use it when you are inside any function call like Plot[]. The shortcut is Ctrl+Shift+Space and you can use it to insert the specific options of the function you are in. This means, in the case of Plot[] and with the cursor inside the braces, after pressing Smart Completion all the options of Plot[] will pop up and you can simply insert them. It's even better, you can just start typing for instance Plot[x,{x,0,1},PR] and when you then press Smart Completion, it will preselect the list of options and show only the ones that match the Camel Humps. This Smart Completion will work in a similar fashion if you are inside Message[], only that it will search your code for messages you have defined and show them as a suggestion.
Finally, sometimes even the clever autocompletion will not show the symbol you have defined anywhere (I want to know if something like this happens!). For those cases, there is an emergency autocompletion which can be invoked with Ctrl+Alt+Space. This one will show you all symbol names you have used anywhere in the file. If nothing helps, this one does.
During coding, nothing is more annoying than taking the hand off the keyboard to use the mouse. There are some keys you should definitely remember because they probably spare you some of those mouse events. First, holding Ctrl down and using left and right on the arrow keys lets you jump expression-wise. If you are inside a deeply nested construct, you can move with Ctrl+[ or Ctrl+] to jump to the beginning or ending of the current scoping expression. You can use Alt+Up/Down to jump between expressions at file-scope, which are usually the single function definitions you made. Holding Ctrl down while using the arrow keys for up and down lets you scroll the document without moving the cursor position. If you want to navigate to a different file tab, you can use Alt and Left/Right.
Finally, with Ctrl+E you get the recent files dialog which lets you move through your current files easily. Ctrl+Shift+E gives you the dialog where the files are sorted by edit time. To switch to the project pane you can simply use Alt+1 and note that all the other panels can be accessed in the same way using the appropriate number. If you want to move back to the editor, just press Esc. The Structure View can be accessed by using Alt+7. Very important is, when you are in the Project View, the Structure View or in one of the Recent Files views, you can not only move around with the arrow keys, you can just start typing to search for a name.
Selecting text works like usual by holding the Shift key and using the arrow keys or page-up and page-down. One incredibly useful feature is the selection of increasing/decreasing expressions which can be invoked with Ctrl+W and Ctrl+Shift+W. The big advantage of this type of selection is that you are often left with valid code if you cut the selection out and your braces are still balanced. Additionally, you can use Ctrl+Shift+] to select until the closing brace of the function you are currently in.
One very nice feature, once you have a selected code block, is the usage of Ctrl+Shift+Up/Down which will move the whole selection block through your code. This is extremely helpful if you want to re-arrange blocks of code.
Live Templates are very short abbreviations for a bigger code construct, for instance, mdl followed by a Tab key will insert a complete Module[{vars}, Null] with template positions vars and Null you can fill in. To move between the template positions, just use Tab. The Live Templates will appear in the autocompletion box too, together with a description of the template. In general, we have tried to stay consistent and template abbreviations are the function names without the vowels: mdl→Module, wth→With, tbl→Table etc. The complete list can be found under Settings → Live Templates → Mathematica. If you don't believe how useful this is, then try usg and start typing the function name.
Another highly used feature of mine is Surround With which lets you surround a select portion of text with all kinds of things. In Mathematica, you often want to surround an expression with another function. This can now be done easily: Use Ctrl+W to select the expression you want to surround and then press Ctrl+Shift+T and then Enter (or 1). With this, it is now even very easy to turn a larger portion of code into an anonymous function.
The Structure View is one of the latest features. In short, it gives you a direct and clean way to see all the definitions in your file. It will extract function definitions, variable assignments, usage messages, setting of options and attributes, setting of tags and upvalues and some more. The Structure View can be turned on with Alt+7 and you have several ways to adjust its behavior. First, you should use the small gear-wheel icon to turn on the Toolbar. There, you can (from left to right) sort all assignments in the structure tree by name or by line number. Furthermore, you can group the entries by the type of the assignment (:=, =, Options, Attributes, etc.) or by name. The other buttons are for expanding and collapsing the group nodes and for synchronizing the Structure View with the cursor position in the file.
In a highly complex source file, the Structure View is very helpful to find definitions. Remember that you can search all visible nodes in the Structure View simply by starting to type. Expanding the nodes can be done with Ctrl++ and collapsing with Ctrl+- which is (again) faster than taking the hand off the keyboard and using the mouse.
Conclusion
What I have described here are the Mathematica-specific features of the IDEA plugin during editing. Keep in mind that IDEA has a lot of more features that are not specific to any language. It has, for instance, a very nice support TODO annotations inside comments or support for different revision control systems like Git. All this can and should be used too.