Wi-fi, smartphone, iPhone questions?

Is the availability if Wi-fi in libraries, restaurants, malls, etc no longer nnecessary since the innovation of smartphones that can pick up the internet just about anywhere?

Another related question: I was in a restaurant today and tried to get online, but a message appeared saying wi-fi access was available, and it interfered with my ability to get online like I normally do. Why did that happen, and how could I have circumvented that problem to get online?

Also: How do smartphones access the internet…via satellite, or radio towers?

I just got my phone about a month ago, so this is all fairly new to me. Thank you