Move the cursor to where you want to edit then press i
Edit as needed, when finished press esc then :wq to save your changes
If you make a mistake press :q! you exit without saving changes, then start again. A cheat sheet for vim commands can be found here: http://www.tuxfiles.org/linuxhelp/vimcheat.html
Now go to another machine, I used windows 7, and install putty. (download here http://www.chiark.greenend.org.uk/~sgtatham/putty/download.html)