Warning: some of the statements below have to be qualified, or are incomplete. Refer to the Emacs documentation (e.g., via C-h i) for the full story.

Generalities:

Here are a few basic terms and abbreviations:

USEFUL FEATURE: in the echo area, when it makes sense (e.g., command names, file names), hitting SPACE or TAB will complete what you typed, or give you a list of possible completions (SPACE will complete at most one word first).
Notice the message in the buffer that contains the possible completions: you can pick your selection from there! Normally mouse-2 is the middle mouse button, mouse-1 the left one, etc.

SUMMARY: one can perform an Emacs command in one of the following ways:

For starters, there are quite a few Emacs commands listed in the menus at the top of an Emacs frame.
Often you are also given the key-combination corresponding to that command.

For example, under the menu "Files", you can find the command "Open File..." The dots suggest that you will have to do more when the command starts. Indeed, you have to give the name of the file - in the echo area - then press RETURN.

But you also see that you can run the same command by typing C-x C-f.

Actually, this key-combination is bound to the Emacs command find-file, which is a piece of Lisp code designed to do what you expect in this situation.

Some Emacs commands are bound to short key sequences. There are many more commands than "simple" key-combinations. But you can invoke a command by using its full name.

To invoke a command by its full name, type M-x, then the name of the command, and finally hit RETURN (notice that all this happens in the echo area).
You can hit RETURN even before entering the whole command name, provided there is only one "immediate" completion.
Example:

Emacs Commands:

Some of the commands are "toggle": invoking them will switch between disabled and enabled.
Some change the information on the "status bar" at the bottom of the frame.
If the command has a standard key-binding, that is given as well.

First, here are a few commands that make Emacs more convenient:

auto-fill-mode (toggle)
Breaks lines automatically, when they become too long.
font-lock-mode (toggle)
Highlights in color various parts of the file, according to their role.
transient-mark-mode (toggle)
Highlights the region, when it is active. The region is the part of the text on which "cut" and "copy" operate.
auto-compression-mode (toggle)
Require Emacs to uncompress file before displaying them. Note that the files are not uncompressed on the disk! You can make changes in the file, and it is compressed when you save it.
Check the Customizing section if you want these features as a default.

Now a few editing commands:

undo, bound to C-_ , C-/, and Undo in the Edit menu.
Undo some previous changes. Repeat this command to undo more changes.
kill-region, bound to C-w, cut, and Cut in the Edit menu.
Kill between point and mark. The text is deleted but saved in the kill ring. The command yank can retrieve it from there.
kill-ring-save, bound to M-w, copy, and Copy in the Edit menu
Save the region as if killed, but don't kill it. That is, "copy".
yank, bound to C-y, and Paste in the Edit menu
Reinsert the last stretch of killed text. More precisely, reinsert the stretch of killed text most recently killed OR yanked.
yank-pop, bound to M-y
Replace just-yanked stretch of killed text with a different stretch. This command is allowed only immediately after a `yank' or a `yank-pop'. At such a time, the region contains a stretch of reinserted previously-killed text. `yank-pop' deletes that text and inserts in its place a different stretch of killed text.
set-mark-command, bound to C-SPACE, C-@, stop, insert
Sets mark at where point is. This allows you to define a region without having to use the mouse: set the mark, then move the point, e.g., using the arrows on the keyboard.

Emacs can do much more than editing. Notice that the menus on top will now include items specific to the new usage. But, we can still use the same editing commands, copy and paste between buffers, etc.!
Here are some of these commands:

shell
Run a Unix shell in the buffer called *shell*.
ediff-files, ediff-directories, etc.
compare and mark differences between files, directories etc. Do M-x ediff TAB to see what else is available.
compose-mail, bound to C-x m, and Send Mail in the Tools menu.
Start composing a mail message to send.
calendar bound to Display Calendar in the Tools menu.
Display a calendar, with a lot of features (check the menus).
doctor
Switch to *doctor* buffer and start giving psychotherapy.