I am admittedly not very good at the game, but whenever I run into an assault drone it seems to kill me almost instantly. I know that if I shoot them enough they'll be destroyed, but I never last that long. Is shooting an assault drone an effective way to destroy it, or should I try something else? Or am I better off just avoiding it until it disappears?
An accurately-thrown EMP grenade will instantly destroy an assault drone if you happen to have one (though most people don't carry one).
That said, destruction is often not as wise as just trying to survive - the key there being awareness and avoidance. The drone dies after one minute or after it takes enough damage - since you always know where it is and can stay out of it's path, you can usually make it out unscathed. Knowing the maps is important - most of the time the player controlling the drone is going to keep to major walkways.
I think Ian's answer is partially incorrect. I use EMP grenades frequently, and I'm a pretty good shot. I've thrown many at assault drones without ever killing one. What I recently found as a much better strategy is using a launcher. Launchers will lock on to assault drones and are often a one hit kill.
The best thing is to have Blind Eye Pro equipped along with either a Javelin or a Stinger Launcher. The Blind Eye Pro perk will lock the launchers on faster and inflict complete devastation on the Assault Drone since the perk increases bullet and launcher damage; I've tested this out by using the perk and a SMAW against an Advanced UAV and it took it out with one missile instead of two. If you use the Javelin it will not fire into the air like it does to destroy aerial killstreaks, it will launch just like a Stinger and go directly after the Assault Drone.