I use vim at work for most of my text editing purposes. Vim is a very powerful editor. However, it comes with a somewhat steep learning curve. The best place to learn it would be the vimtutor program. It's got a pretty good "for dummies" approach which is useful if you haven't used an editor even remotely powerful as this one before. If you're short of time, the best way to learn it is to just jump in and start using it. Since it's been around a long time, it has pretty much most features a developer needs. So if you have a need and don't know how to get it done with vim, then use this site.
To start off using vim, let me give you a very quick and dirty first few steps. To open a file, simply use vim <filename>. You most likely have vi pointing to vim. Plus, most of these commands work with other vi's anyway. So I'll be using vi instead of vim henceforth.
Now that you have a file open, you would like to enter some stuff in? The basic thing you need to know about vi is that it has many modes. From the default mode(command mode) you need to press 'i' to enter the insert mode. In this mode you can type stuff in and it will show up on the screen. The command mode is where you issue commands to vi (But you can't enter text). To go back from the insert mode to command mode press the escape key.
Now that you've typed some random stuff in, escape to the command mode. Now try pressing ':w' i.e. (shift+;)w. W is for "write". This saves the file and leaves you still in the editor. If you want to get out of the editor, you type in one more command 'q' so it becomes ':wq'. In case you want to get out of it without saving, use ':q!'. The other way to save and quit is to go to the command mode and hit shift+z+z.
With vi, the traditional way to navigate (in command mode) is to use l,k,j,h keys(right,up,down,left). That's like the a,w,s,d in gaming. It's a timesaver for people who are comfortable with it. Vim allows you to use the arrow keys as well, which is a real convenience.
That covers the bare minimum basics for us.
Once you start using vi you start looking for ways to accomplish tasks using it. There is a whole lot of material out there in the World Wide Web which will help you out. Google is your friend, use it :)
I will tell you some of the frequently used options/shortcuts I use with vim.
Going to a specific line : Type in ':25' to go to line number 25 (in command mode). You can open a file at a specific line by typing
$ vi +25 FILENAME
This will open the file and take you to line 25. Now that you're there, perhaps you would like to delete the next 5 lines? No problem - just hit d5d. The 5 in the middle says 5 and dd is for deleting lines. You will find this a common pattern with most vi commands. Just hitting dd will remove a single line. Similarly you can do d5w to remove 5 words and dw to remove a single word.
Copy paste is a walk in the park. Hit y7y to copy 7 lines. Now press p to paste. That's it. Cut paste? Hit d7d and then press p to paste after navigating to where you want to paste it.
The other approach to doing all this is to send a command. Hit ':201,225y' to copy lines from 201 to 225. Similarly ':201,225d' deletes lines from 201 to 225. To jump to a line you can use ':25'. This jumps to line number 25.
One useful shortcut when you don't want to remember the line numbers is the use of markers. I can mark any 2 lines , say line numbers 201 and 225, and perform an action on them by using the name of the markers. Go to line number 201 and hit 'ma'. This will mark that line with the marker 'a'. Do the same with line 225 and mark it with 'b'. Instead of uisng ':201,225d' to delete lines 201 to 225, now you can use the a and b markers to delete the lines thus - ":'a,'bd".
That was just a small snapshot of the commands that I use on vi. You can use a lot other features, like for instance macros, and enhance your editing experience. This should help you get started though. Let me know if this has been useful. Good luck and happy hacking :)