Just like armor, these are very simple to level up, but take a lot of time. You need to do damage against enemies with either weapon type in order to increase the weapon skill. The stronger the enemy, the faster your skill will increase. When it comes to the higher levels, this will be important as you will find that killing low level enemies, does not help you much.
One thing people like to do is attack the horse that you obtain from the Dark Brotherhood storyline. This horse is called Shadowmere and has a crazy high amount of health. He can also heal incredibly fast, meaning you can just keep attacking this horse over and over and it will never die. This will definitely help you increase your weapon skills without having to find high level enemies or risk getting killed.
Another method you can use to increase your weapon skill is to conjure some creature using a spell. Once it has spawned, attack it and keep attacking until it is dead. This will also help you increase your conjuration magic skill.