You are right, Frankfurt (M) Flughafen is the Airport railway station.
It is very easy to get there, from Terminal 1, just 10 min walking, just follow the signs.
Normally, if you buy your tickets online at bahn.de 2-3 weeks before your trip, you can get a 29€ ticket.
The prices are depending on the train type:
ICE: (InterCityExpress) very fast train, less stops, train restaurant -> pricy
IC: (InterCity) Regional Train, more stops, train bistro
RE: (Regional Express) very slow train, many stops in small cities -> less pricy
If you book online you can pay with a Credit Card. You have to personalize your ticket (name, age, telephone number etc). You have to verify your identity with a creditcard or ID.
You have to print your online ticket. The printed ticket is as valid as the "official" ticket from the counter/ticket machine. You have to bring the ticket and the document you decided to identify with (Creditcards, Maestro, ID, etc).
They often are accepting online tickets on smartphones, iPad, etc.
If you buy your ticket at the station, it will be 3 times pricier.
If you don't want to plan for 2 weeks, there is a last-minute option: ltur.de
On this site you can find 27€ tickets from 1 day to 1 week before your trip. If the tickets are gone, they are gone. This website is using the same interface as bahn.de