What are current solutions to switch to Internet-based TV and movies (see details)?

We’re doing Dish Network on an HDTV that offers 1080i, 720p, 480p & 480i settings as well as on an old school CRT TV. We think we want to switch to something along the lines of Roku + Netflix.

What’s your setup and do you recommend it or something else?

What speed Internet connection will we need to ensure stuff streams well?

Any trade offs worth mentioning (live sports, e.g.)?

Thanks!