Congratulations on your new 'home'! It sounds like you're pretty 'gung-ho' about fixing the ol' place up. But before you do, you want to think about something.
I'm making a BIG assumption here in that you're RENTING your apartment. Assuming that, you'll want to get the Landlord's permission before doing ANY type of work. In short, you could end up in court for who-knows-what-reason. I'll take your word on it that the carpet is old and in need of replacement. But, for one, if you get hurt, technically, you could sue the Landlord then he can counter-sue you. Forget about sueing him for breaking a leg during the replacement job you want to do. Think about sueing him for having old carpet that made you slip and fall and bang your head BECAUSE the carpet is old and in need of replacement. Go to him with that IF that's the case. THEN you can talk to him about doing the job. But don't be surprised if he asks you sign a Waiver.
But, if you insist, go to The Carpet and Rug Institute for Installation Instructions. I'm not sure if they 'do stair' but it's a start. If they don't, e-mail them.