I've been planning for a 4K upgrade later this year (so far my 9y/o 8000 series Samsung 1080P refuses to die
) so I've been doing a lot of research. According to Netflix, for 4K Ultra your internet connection needs to be able to maintain a steady 25Mb/s, more if you plan to support multiple TVs streaming at the same time. I can't speak to wireless because I ran Ethernet to all but my laptop and tablets, those run off 1200Mb/s 802.11ac. My 4K laptop does play 4K Youtube videos pretty well over my WiFi.
https://help.netflix.com/en/node/13444
I have played with several media players and unless you're particularly enamored with FireTV's interface (I hate it) I'd look for other solutions. I find Roku's interface much more user-friendly and the latest flagship
Roku Ultra is supposed to support all of the latest 4K implementations. Unfortunately, the 5-second delay between the start of a video and the start of the accompanying audio drives me nuts (but less so than the Fire's interface), although the Roku's audio/video sync is perfectly fine. I'm looking forward to trying out an Nvidia Shield TV.