We'd love to get rid of Time Warner cable and are looking into Roku. Are you able to watch any TV shows with this? What about local channels?

Would we need a converter box and rabbit ears as well?
Gotta have American Idol

Any info is greatly appreciated. Thanks!
As much as people would love to get rid of their cable and switch to an interest-based delivery channel, such as Roku, there is much to consider...
Yes, you can watch some TV programs, but not everything - especially, "live" programs such as "American Idol" and other vote-centric shows.
Are you a big CBS watcher? Forget the "box", as CBS has no agreements with internet content providers, such as Hulu Plus and Netflix to deliver their TV programming (Hulu is a joint venture of NBC/Universal, FOX, and Disney/ABC).
Like live sports? You won't find them on the "box" unless you are willing to pay extra. A good example is MLB.TV, which is available on Roku for an extra $120 that covers the entire season from April-October. The only bad thing is you can't watch your home team (meaning if one lives in South Jersey, they won't be able to watch their Phillies games, as they are blacked-out by MLB).
You may also need to boost your internet speed, as some of the "entry-level" plans cannot support streaming video very well (you will know this with frequent interruptions to "refresh" the signal).
I'm not totally against the box, as I have a Blu-ray player that "streams" Hulu, Netflix, and others (I like Hulu, because I can revisit some old programs I haven't seen in many years). In fact, I would recommend if you want to test the waters for yourself, to buy a Blu-ray player that is streaming-capable (most can be had for around $100). If you decide that Hulu isn't your cup of tea, you still have a Blu-ray player that you can watch all those Disney Blu-ray movies you saved 10 bucks on...
As for local channels, they are not available on any streaming services - the best way to get your local stuff is to get a good outdoor antenna.
Digital converter boxes are not needed for most TVs manufactured from 2006 forward (check your TV set owner's manual).
Hope this helps.