How do I clear possible browser malware on an iPad?

I am new to iOS, so please bear with me if my question seems a bit ignorant…

I recently purchased an iPad Mini 3 (LTE model)—it's running iOS 8.3.

Today I clicked on a seemingly legitimate website using Safari—I happened to be using private browsing mode at the time—and apparently hit some type of phishing attack. My iPad's screen immediately turned grey, and a pop-up appeared congratulating me for winning a prize and asking me to click "okay" to confirm the prize. The pop-up also correctly identified my mobile service provider. I did not touch the pop-up. Instead, I followed my first instinct, double-clicked the iPad's home button, and tried to close Safari by swiping it upwards. When I reopened Safari, the grey screen and the pop-up were still present. I then went to my settings app and pushed the "Clear History and Website Data" button; this action seemingly solved the problem. I subsequently reported the offending website to Google as a phishing site.

Two things concern me about this incident:

First, how was this malware able to determine the identity of my mobile ISP?

Second, is there any danger that additional malware or spyware was installed into either my browser or my iPad, even though I did not click "okay"?

What should be my next course of action?

Is it necessary to change the password connected to my Apple ID and the two other email accounts that are installed on this tablet?

Any advice would be greatly appreciated…