To increase your speed:
1. Turn off explosions in your preferences.
2. For those of you with a track pad, get a mouse.
3. Use the control/alt key when attacking multiple countries so that all of your armies attack and advance. When using a mouse, make sure you right click as well, otherwise only two armies will advance which will slow you down.
4. Turn off Army Graphics in your preferences
When you kill another player, you get their cards. If their total cards plus yours equals five, you cash them in for more armies. If you alt/control and left click the mouse (or just left click the mouse) when you take their last country/kill their last army, you can add your attacking armies minus two to the cashed armies you gain and distribute them on the board where you want.