Once you have org installed, hit "Control-x f" to open/create a file, and name one "orgtest.org". Type "* first header" -- if it doesn't turn blue, type "alt-x org-mode" to turn on org-mode. Now type "alt-enter", then hit tab a few times -- it should change the number of stars (level of indentation). Leave it with two stars. Type "take dog to park" on that line, now hit "control-c control-t" several times. It should cycle through a todo/done tag. Hit a regular enter this time, and type "remember your frisbee". Type "control-c control-s"; a calender should pop up, click on tomorrow's date. Now type shift-tab a few times and watch the outline fold/unfold.
Note that the date line and TODO tag are plain text, as is the entire .org file. It could be helpful to think of org-mode as a code editor and interpreter for org markup.
I believe the above should take you to the point where you can follow org-mode tutorials and reference material, although if you decide to really dive in you should probably take some time to learn emacs as well.