While many people believe in the power of installing internet filters or whatever more, this is actually something that you definitely should NOT do! And yes, many people will call me crazy but as a parent it's a way to tell your child: "I don't trust what you do online!" which is something children don't want to hear. Only install a filter if the child wants one too and teach your child how to disable this filter again if the child ever needs to get around it.
The best thing you can do is the same as you can do when you let your child walk on the streets all by themselves: educate them as best as you can and don't try to avoid any topics that they might come up with, no matter how uncomfortable such topics will make you. (But feel free to explain that you feel uncomfortable about that topic.)
First of all, filters are unreliable. China has created the Big Firewall of China hoping to control the contents of the Internet within it's borders. They're failing miserably, no matter what they try because there are plenty of options to go around it. Myanmar even closed off all Internet connections within it's borders simply because of the civil disorder. They didn't want to have any images of political abuse leaving the country. They failed too, although they did manage to keep a lot hidden from the outside world.
If you tell your child not to visit porn sites or whatever, then your child can just obey or disobey your wish. When you install a filter then you will just encourage your child to find a way around this filter, if they really want to find porn online. A filter isn't going to stop them! Maybe they'll just go to a friends place, where the friend has no filter and thus unlimited access. Or maybe they'll visit a library, if the library has no filters. Or they find a way to disable your filter. Anyway, children are cunning enough to get what they want, if they really want to.
Also keep in mind that a filter might block too much, or too little. For example, this post might be blocked simply because it uses the word "porn". If that's no reason to block this post, words like "penis", "vagina", intercourse", "condom", "viagra" and "aids" might be enough to block these posts, even though these words would be common words for a medical site or even Wikipedia... (Btw, if your filter blocks this post, please raise your hand!)
But how do children respond to those "bad sites"? Well, if they were looking for it on purpose, they'll be happy. But when they're just browsing the Internet in search for something else, they just behave like most adult Internet users: take a peek out of curiosity and then just ignore it. Children who discover porn sites or sites containing pictures of mutilations and worse material might be upset at first but will soon learn to ignore it. By educating them how to ignore such sites and by teaching them what to do when they discover a bad site.
And finally, talk with your child about installing a filter and explain to them how they can disable it! This tells them that you trust them to do whatever they like and that the filter is there to help them, not to block them. If the filter is voluntary, your child will much more appreciate you for it. (And they won't try to disable it simply because they don't have to.)
People who try to educate children by restricting their options simply forget that if a child wants to do something, there isn't much that will stop them from doing so. Children aren't just home enough to control them 24/7. School will bring them in an environment where they can learn all the dirty tricks. And visiting friends will give them enough possibilities to do things without your knowledge. A filter won't stop them. At most, it will just annoy them.
The biggest problem with filters is that they give a false sense of security. People start to believe in a piece of software to do their parenting for them. This makes absolutely no sense! There are people who just install a filter, thinking their child is safe, while their child just goes visiting a friends place just to look at everything you've forbidden.
Filter software only has an advantage for those who create it. They earn Big Bucks by making parents paranoid about the dangers of the Internet.