This is all assuming your iOS is reasonably up to date and/or don't have it jailbroken
The apps are not necessarily secure, as malicious apps have made it onto the store, but much more likely to be secure if all apps come from the app store and the apps have been vetted. I would trust the iOS closed system without AV more than an open system like a PC or Mac with AV.
As for web browsing, the fact that you are not a root user, the way the OS works, and the lack of 3rd party plugins means it would be very difficult for a web based exploit to do anything of real concern.
In short, don't worry about it.
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