Closely watching legislation progress through the House can be difficult, especially given the discretion the Speaker and the Rules Committee have in scheduling and controlling the floor.
The House Rules Committee is making legislation a little easier to follow online, by voluntarily posting updates of special rules, as they are approved, to their website in a structured format. This means that you can receive updates on newly approved special rules that structure floor consideration for impending bills. They’re also offering a feed of amendments submitted to the committee.
The only way to get faster updates would be to attend the committee hearing.