I find that for many people learning League of Legends, what to do after the first tower in your lane (allied or enemy) goes down presents a challenge. There is some good advice in this thread.
Here's my take -
If you destroy the enemy tower - Move to a nearby lane and try for a gank. Team up with your jungler, if you have one, for even better results. Keep an eye on your lane so that the enemy doesn't get to immediately push down your tower for free, but add another roamer to the list of things that the enemy has to worry about. If you can grab a few kills or assists you'll get some nice money AND help put other lanes ahead.
Beware of trying to push/cs to the next enemy tower and deeper into enemy territory without enough wards. When you are in that far it becomes easy for the enemy to gank you by walking through their own relatively safe jungle.
If your lane opponent continues to push while you are gone, return every once and a while to clean up the minion wave and claim the CS while stopping their advance. If they push really far, bring a friend and take them down.
If the enemy takes your tower - This situation can actually work in your advantage if you are a good farmer. Carefully last hit to keep the minions balanced in the lane, and if the enemy laner leaves to gank quickly clear the wave and wait for their turret to bring the wave back to you.
If you can keep the minions on your side of the map, you'll have a safer farm and it will be dangerous for the enemy to continue farming. Your team will have to manage the threat of having another enemy that can be roaming around, but if they can avoid getting killed you can lower that enemy's gold flow a bit!
In the true mid game, when teams band together and start fighting each other, you need to be careful about farming lanes. Not just because you might get ganked if you go to bot to clear a wave, but also because teams will being targeting objectives like Dragon, Baron, and your remaining towers.
The advice to ward that others have given is really good. Also keep in mind that if you go bot and the enemy team sees you there farming there alone, they might decide to jump the rest of your team at mid and have a 4v5 fight or snag Baron when you aren't there to help drive them away. Communication with your team is key. Let them know that you're clearing a lane and that they shouldn't fight without you.
On the other hand, you can also use these tactics against enemy players. Map vision in the mid game has a lot more uses than just preventing ganks!