The most common entry point into learning vim is vimtutor. You can start learning vim by simply invoking vimtutor from the command line. Vimtutor launches vim and opens the vimtutor text file which contains instructions for getting familiar with vim's basic functionality (modes, navigation, editing, search etc.). Finishing this tutorial will not take too much time and it will teach you the basics of how vim operates.
The power of both vim and emacs comes from customization. While vimtutor will teach you basics of vim's editing capabilities, most people find it handy to add some custom commands. If you want to learn more about this, a good resource for starting would be the Learn Vimscript the Hard Way book.
If all of this looks too bland for you, there is a web-based game called VIM Adventures, that serves as a tutorial for learning to control vim.
Once you're feeling comfortable with basic Vim usage, it's time to add some plugins to make your life easier. Click here to learn how to do that.