What's the best way to connect Nintendo Gamecube to TV?

While component will allow you to see 480p in the games that support them, 480p is still the same resolution your old SDTVs carried. The quality of 480i or 480p on an HDTV is atrocious. Component cables won't change that at all unfortunately. Technically, it will give you the best possible quality due to the progressive scan, but it won't be close to HD quality.