I'm speaking as a layman here. There are a lot of people on this forum with a lot more technical knowledge than me but I'll give this a whack.
Just deleting your internet history wouldn't be enough because your service provider can still see what sites you visited and the information you sent and received. You can download TOR or similar browsers that the pro-privacy community have been nice enough to design for us less tech savvy peeps. Even then though, I don't know enough to say that this will make you 100% safe.
You could use a VPN (whatever that is), guaranteeing you even more security. As far as I know they cost money and unless you are a hardcore paedophile or terrorist this might be overkill.
The safest way I could think of would be to use a stolen laptop or Android burner phone in an internet cafe that you never frequent. This is the safest method I can think of. It's also idiot proof - get a (clean) web capable device and go to a WiFi hotspot then ditch the phone. I don't think you can beat that. Just look out for witnesses and CCTV dressing accordingly.
If you're looking for a way to be totally anonymous then forget doing so from your home internet connection.