Depending on the service, you need between 10 and 25 megabit-per-second internet to stream 4K video.
That shouldn't be a problem for most people in the US. The average internet download speed in the US in 2017 was 64.17 Mbps, according to the internet speed testing site, speedtest.net. That's more than enough for 4K streaming.
For downloading movies and TV shows you bought, faster internet speeds means faster downloads and less time between the moment you click "buy" and actually start watching.
As for WiFi routers, if you're still using an old "N" router, it might be time to get something newer. You can usually tell what standard of WiFi router you have with the model number printed on the router itself. If there's an N in the model number, it means i'ts pretty old and could be bottlenecking your internet speeds.
For newer routers, look for those with the "AC" WiFi standard.