While completely blocking websites on cell phones and gaming consoles can be extremely tricky, you certainly can easily make a few changes to make sure that your child doesn't accidentally stumble on something even if he isn't actually looking for it.
You don't even need a smartphone to access the Internet - any underachieving feature phone will allow a reasonable amount of Internet access. The problem with phones of course is that they are used outside of the home. It can be very difficult to monitor what goes on on a phone's Internet browser. Now certainly, your child isn’t going to accept a phone that doesn't access the Internet. That would brand him a loser in his school. What you could do instead is to apply with the carrier, Verizon, AT&T or anyone else, to activate content filtering on your child’s phone. This only works for feature phones though and not for smart phones. On smart phones, you'll have to get a browser like the K9 browser that allows parental restrictions. And in case your child tries to install another browser, you have to turn that facility off in the Restrictions menu.
On both the Sony PlayStation 3 and the Wii, you can just switch the browser off altogether. You don’t have to worry about how that could cripple the machines in their gaming abilities. The browsers are just incidental add-ons. If you don't want to do that, on the PlayStation 3, you get to subscribe to Trend Micro's parental blocking service that costs $20 a year.