Updated: 10.06.2015
Seriously, the mouse is evil. Don't use it, it's one of Douglas Engelbarts horrible creations, even worse than his keyboard which had only 5 keys.
1. Use a tiling window manager and do not enable mouse support. Good examples are ratpoison and bspwm.
2. Use urxvt as your terminal and install urxvt-perls for easy copy and link opening.
3. Use the (n)vi(m) or Emacs editors, and don't even think about turning on mouse funcionality.
4. Use the a command line based email client like mutt or alpine if you'd like something simpler.
5. Replace your file manager with Midnight Commander or prefferably Ranger if you want vi controls. Gnome-commander if you fancy a gui.
6. Install vimperator/pentadactyl on firefox or make the leap and use the wonderful vimb which goes perfectly with a tiling wm. (chrome + vimium is crap)
7 Find a nice command line music player. Cmus is a good example, but you can go for an mpd setup and use vimus or ncmpcpp.
8 Play videos with mpv.
9. Join irc channels with a free irc client: irssi and weechat are favourites.
10. Check your feeds with newsbeuter.
11. Downlod torrents with rtorrent.
12. Manage your volume with alsamixer.
13. Organise your calender with khal.
14. Multiplex your terminals with GNU screen, or tmux.
15. Open pdf files with mupdf or alternatively apvlv (slower but more features), and you'll feel right at home with vi controls.
16. Next time you join a wifi network, please use the alternative for your network manager such as nmtui, wicd-curses, wifi-menu or wpa_supplicant if you want to be in full control.
Some nifty usability ones to spice up your set up include: ratmenu, dmenu, lemonbar or conky.
These programs are probably going to be good for 99% of the things you do, and they are all the best in what they do, partially because they don't make use of the mouse. They also work great together since:
a. You can use the window manager bindings to execute cmus-remote or mpc to control your music in cmus or the mpd respectively.
b. The programs can easil be scripted. You can write or find a perl script that runs a system notification program when you get a new message in irssi or weechat.
c. The tiling window managers bspwm and ratpoison can be controlled bspc and ratpoison -c so you can make the integration between all the programs and your window manager seemles as you can have any program trigger any other.
Sounds amazing? Check these out and get some inspiration on the things you can do:
Angelic Sedition's Awesome Dotfiles
All my Dotfiles with Configs and more
And if you want a good fresh start with some nice programs check out the sick Ratpoison version of Salix:
Distribution with 0% mouse usage
P.P No mouse was used to make this post.
Posted without the aid of a rodent.