What is the best connection type to insure the clearest picture?

There's absolutely NO DOUBT that wired connections are more stable than Wifi especially in environments where there is substantial channel congestion. I've relied on a wired connection for years and have yet to have any problems with resolution, updates or other functions. If you have Ethernet ports near your Roku player or TV, I suggest you connect with them rather than WiFi. I use a power-line network which if unfamiliar, utilizes a home's electrical circuits to deliver data via special adapters. Hope that helps