How to push streamed music from iPad to TV
The easiest way imho would be to buy an AppleTV2. It's cheap, and "it just works".

Edit: For completeness sake, I'll just clarify what you can expect from the AppleTV2. Since it is an AirPlay compatible reciever for both audio and video, you can stream music or videos being played on your iPad (or iPhone/iTunes for that matter) directly to your TV. On your computer, you can even stream any audio source through software like Airfoil. Starting with iOS5, AirPlay even allows you to mirror your iPad on the television set.

Additionally, if you enable Home Sharing in iTunes on a connected computer, your iPad can serve as a remote to play content (audio/video) in your iTunes library directly on the TV. For this usecase the content doesn't even need to be on the iPad. You'll just need the Remote app by Apple, available for free in the App Store. The only downside here is that your computer (and iTunes) need to be powered up to access the media.