I'm not an Apple person, but I try to be objective and regularly use iOS, Android, and Windows Phone. My opinion is you are better off sticking with iPad for tablets (unless you want more power in the form of a tablet-laptop hybrid, in which case go Surface Pro 3 or Surface 3). The hardware is class-leading and it has better tablet optimized apps than anything else... the exception being the times when there is only an iPhone version of an app. Running a scaled-up app in a window is uglier than how Android handles non-tablet versions of apps.
For phone, Android is much more flexible and powerful platform, and you likely won't regret switching to it if you get a flagship level phone. iOS feels very much behind on design and customization, and it doesn't use screen real estate on larger as effectively as Android can. iOS is still mostly a static icon grid, which is perfectly fine though if all you care about is apps and don't spend time on the home screen.
Apple TV is good for Netflix and AirPlay, but Roku is the most complete solution in the set top streaming solution market. Chromecast is what I use, though, and I haven't had any issues with it. Cheap and easy to use and you can cast from iOS devices as well as from Chrome or Android devices.
If you want to go all Chrome/Android it's certainly possible to get a good experience you get a Chromecast, Nexus or Samsung tablet, and a high end phone like the GS6, G4, or the next Nexus. I don't like ecosystem lock-in, though, so I jump around.
I do have a bias in favor of Windows and Windows Mobile (Windows Phone), but Windows 10 isn't out yet, and it still needs a few months at least after that to get the bugs worked out, I would assume. Once Windows 10 is available and more hardware starts coming, it might be the best platform in terms of a unified experience across all devices.