The cheapest way to travel from Frankfurt to Hannover is typically by using regional trains. Here are the steps you can follow:

1. Regional Trains: Take a regional train from Frankfurt Hauptbahnhof (main train station) to Hannover Hauptbahnhof. Regional trains are slower than high-speed trains but are generally more affordable. The journey duration can vary depending on the specific train and route, but it usually takes around 3 to 4 hours.
2. Ticket Options: To get the best fares, consider purchasing a ticket for a specific regional train rather than a flexible ticket. Look for Sparpreis or Quer-durchs-Land tickets, which offer discounted rates for specific train connections. These tickets are often cheaper when booked in advance.
3. Online Booking: Use the Deutsche Bahn website (http://www.bahn.com) or their mobile app to check the train schedules, fares, and availability. You can book your tickets online and either print them or have them on your mobile device.
4. Consider Connection Times: When selecting your train, pay attention to the connection times. Opting for trains with shorter layovers between transfers can help minimize the total travel time.
5. Alternative Options: If you are more concerned about cost rather than travel time, you can also explore bus options. Bus companies like FlixBus and Eurolines operate routes between Frankfurt and Hannover, offering budget-friendly fares. However, keep in mind that bus journeys tend to take longer compared to train travel.

It’s important to check the most up-to-date information on train schedules, fares, and any travel restrictions before your trip. The prices and availability can vary based on the time of booking, so it’s advisable to plan and book your tickets in advance to secure the best rates.