My kids are now in their 20's so my advice is less about products than things to consider (although I can second the comments on Net Nanny - I had to completely uninstall it with tech support help)
Sometimes children can come across inappropriate content inadvertently (hint: be very very careful when searching for info on how to tie knots or anything about Barbie dolls) and there used to be search sites that did some of that filtering up front based on an age range (when mine were young, it was Yahooligans).
Computers should be in an open space like the living room, kitchen, great room - not their bedrooms - where you as the parent can keep an eye on things. Don't stand behind them, just keep an eye on things. A logging tool is very useful, and checking browser history. If they have other devices like cell phones, the chargers need to be in a central location (not in the bedrooms).
We had an issue with cyber-bullying at one point, and another time when someone with questionable motives was chatting with one child. As a parent, you need to encourage your child to talk to you about what they're reading and seeing on the internet. The cyber-bullying was the hardest one to cope with - the schools don't do a good job of addressing it, the other parents deny that their child could be behind it, and you have no real evidence of anything.
Is it hard? Yes - when they had homework to do on the computer, I turned the TV off so I could keep an eye on things and talk to them. They'll complain about not having the computer or cell phone in their room and you'll be the bad guy who enforces the rule (which also applies to you, mom and dad).