If you have not done something to install an app from outside the App Store, you can't be infected with any known kind of malware or adware. If you have done this, you may be infected, and will need to restore the phone to factory defaults without restoring from a backup.
It sounds like your primary problem involves browsing the web and getting either pop-ups or redirects that open other apps (I'm guessing the App Store). If that is correct, you may be on a wireless network that has been compromised. See:
How can I troubleshoot my wireless router or modem with my Mac?
(Fair disclosure: I am affiliated with Malwarebytes, whose site I am linking to above.)