There are 3 different ways to sign up for YouTube Red: from YouTube.com or the Android app, from Google Play Music and from YouTube's iOS app. If you use in-app purchases on iOS, you'll spend 30% more every month ($12.99 vs $9.99 regular pricing), so it's not a good idea to do that.
The desktop site has a new logo:
Here are some screenshots from the iPad app:
"With a YouTube Red membership, you’ll experience YouTube without video ads, be able to save videos to watch offline, and play videos in the background on your mobile device, all for $9.99 a month. And just like with our advertising revenue, the majority of revenue we get from YouTube Red memberships will go to our creator community," mentions YouTube's blog.
YouTube Red is free for Google Play Music subscribers and there's a free 30-day trial for US users. For now, YouTube Red is limited to the US and there are some additional restrictions: "if you leave the U.S., you won't be able to save videos offline, videos won't play in the background, and you will see ads. Any videos that you've saved offline before leaving the U.S., will continue to be available offline for 30 days."