Use cases of org-mode as a scientific productivity tool for academics without programming needs
I use Org-mode and AUCTeX (Emacs LaTeX package), to do all three tasks you outline. I have an Org folder that I sync across machines using Dropbox, which I find to be a simple solution for someone who does not use version control on a regular basis.
Organization
I separate my tasks into broad groups with each group getting its own .org file. For example, I have .org files for administrative tasks, journal publications, service, research topics, and any major projects that I am currently working on.
The structure of an .org file is relatively simple, for example a file to track journal submissions, revisions, etc. may look something like this:
* Initial Submissions
* Accepted
* Rejected
* Revisions
* Book Chapters
Org-mode uses asterisks to denote levels of headings, and <Tab> to
fold and unfold the headings. So expanding the revisions heading would
lead to:
* Revisions
** Paper 1
** Paper 2
DEADLINE: <2012-05-04 Fri>
You can set deadlines for any task by pressing C-c C-d, which will
generate the the DEADLINE: line you see above. Setting a deadline for
a task will make the task show up in the agenda view (accessed through
C-c a a), which is my main project planning tool for day to day
work inside of Org-mode.
You can also track the time you spend on tasks with C-c C-x C-i,
which will clock you in to a task and C-c C-x C-o, which clocks you
out. The tracked time will show up in the agenda view and can be
useful for project planning or reporting. You can also generate
separate standalone tables inside your .org files if you prefer a bit
more customization.
All of this can be done with a vanilla Org-mode install and no customization. I have my tasks set as multi-state TODO lists that I can cycle from TODO->STARTED->WAITING->DONE->CANCELED. I have my keywords set in my .emacs configuration file with the following:
(setq org-todo-keywords
'((sequence "TODO" "STARTED" "WAITING" "|" "DONE" "CANCELED")))
The "|" separates in process keywords from finished state keywords.
If you are looking for more elaborate reports, such as Gantt charts,
my answer to this question briefly discusses some of the options
available.
Outlining
For outlining and writing long documents, you can just create a new .org file and outline using the * heading approach. Org-mode makes it easy to move headings around if you want to restructure your document at any stage. For example, if you had this outline:
* Intro
* Part 2
** Part 2a
** Part 2b
* Part 1
** Part 1a
** Part 1b
You realize that Part 1 should really come before Part 2 so you move
the cursor to the Part 2 heading and press C-<down arrow>, and Part
2 and all of its subheadings will move to the proper position.
* Intro
* Part 1
** Part 1a
** Part 1b
* Part 2
** Part 2a
** Part 2b
Depending on your needs, writing a paper based on the outline can be done in much the same way. Org-mode has support for LaTeX, both for inline fragments and for environments. Since you mention LyX, I would imagine the transition to stand-alone LaTeX should not be too onerous. The Org-mode LaTeX export does a fairly good job but if you have a document with a significant amount of LaTeX syntax, it may be better to just write the draft in LaTeX using AUCTeX, but this is beyond the scope of the question.
Reference Management
I use a combination of Org-mode and RefTeX (available with AUCTeX) to manage my references and to make notes. As mentioned in John Moeller's answer this takes some non-trivial configuration. I used this setup almost verbatim to start my reference management, and I have found that it works well. This link was inspired by the same setup and may be useful for both reference management and writing drafts in Org-mode that contain extensive references.
I start with a master .bib file, that contains the bibliographic
material for each reference. After adding updating the .bib file, I
use C-c ) to insert a new heading into my notes.org file. The
customization will generate a heading with the title of the paper and
a link to the PDF of the paper. For any notes I take on the paper, I
can use the rest of Org-mode's abilities to either have multi-heading
outline style notes as subheadings, or just write paragraphs separated
by a blank line. The end result is an .org file with headings for
papers, textbooks, etc. and subheadings for each paper with links to PDFs and all my notes in a single file.
Tips for Starting Out
There are a few ways to help smooth the way to working with Emacs, Org-mode, and AUCTeX.
Install Emacs 24 pretest instead of Emacs 23. Emacs 24 has package management included in the vanilla install which makes it much easier to add packages without a lot of programming experience. It also has Org-mode included in the default install. This link gives instructions for a variety of operating systems. I have been using it for awhile now and I have found it to be very stable.
Go through the Emacs tutorial, accessed via C-h t. This will give you the basics of navigating using the Emacs keys. It will likely take some getting used to, especially how Emacs handles selection, cutting, and pasting. This will likely be the biggest hurdle if you are used to the cutting/pasting/navigation in word processors.
Keep this reference card handy. It has nearly all of the commands that you will use on a daily basis.
For Org-mode specifically, look through the manual, but more importantly look at the tutorials. Specifically the general introductions and the power users describe their setup sections (the first two sections linked above). These tutorials will highlight the customizations made to the initialization file (.emacs) for these users. Even without elisp experience, you should be able to find something close to your desired workflow and be able to modify it with some trial and error.
When I started with Emacs and Org-mode I had very little experience with Emacs. A vanilla Org-mode install with no customization is still a powerful tool. As you get more comfortable with working in Org-mode you can start to work on customization. Even with very little interest in programming there is a significant enough user base that someone may have already done something close to what you are looking for.
After I got comfortable with Org-mode, I started using Bernt Hansen's set-up with no changes. It is a bit intimidating on the whole as he has some extensive customizations, but he documents them well and explains almost everything he does. Then after using it for awhile, I was able to modify the initialization to something that better suited my workflow. It took some trial and error and a bit of extra time on the learning side, but I believe that it has payed off in the long run.
Once you are comfortable with Emacs, I would also recommend the Emacs wiki. It has some descriptions of useful packages, some discussion, and even some configuration suggestions to help build up your initialization file. If you ever get to the point in your setup where you think, "I wish I could do XXX", the odds are someone else has written a package that covers what you need.
I am an academic [history] who adopted org-mode about 18 months ago. Frankly, I'm puzzled by all the warnings about the steepness of the emacs learning curve. For a newcomer who doesn't do a lot of command-line work, the hardest part for me was configuring emacs on my Win and Ubuntu machines. But there are lots of resources and tutorials out there, many of them accessible from orgmode.org.
I started out only using emacs for org-mode. After watching a few screencasts I knew enough to start outlining. Gradually i've been using emacs for more and more tasks outside org-mode, though that remains my main use for the editor. I've been gradually increasing the complexity of my workflow over the last year, adapting bits of the various GTD setups linked to the org-mode.org. I am not a programmer and had only started working with a Linux machine a few months before getting into org-mode.
In my opinion, one of the obstacles to greater adoption of org-mode is that people see the amazing workflows set-up by gurus and assume they need to use emacs at that level. My opinion: for writing, organizing and work-flow, you can get 80% of the ultimate value of org-mode in about 20 minutes of instruction.
I use org-mode as an grad student (computer science), and primarily use it for throwing together fast documents. It isn't great for papers/articles, but it is great for homework and notes, because it uses markdown for formatting. There is very little to learn here that can't be found in the manual. See the sections on exporting/publishing, and pay attention to the parts on LaTeX.
I used to use org-mode for scheduling and it was great (I went on vacation and lost interest in tracking all of my time). It's great for tracking how much time you spend on projects and doesn't need much configuration up front.
I also used it for collecting references, but that took some nontrivial configuration. It wasn't hard for me because I'm used to programming, but your mileage may vary.