Frankfurt to Hannover by train

DB is the national railway operator in Germany, so its website can be used for booking journeys by German express trains; the ICE and IC trains and it doesn't charge booking fees.
It also sells tickets for journeys by direct trains on all international routes from Germany regardless of whether DB is operating the train service.

It also sells an extensive range of end-to-end journeys which involve making connections both within Germany and in neighboring countries, but journeys between Germany and Britain cannot be booked on DB.

A key feature of DB website worth keeping mind is that it offers 1st class ticket purchasers complimentary seat reservations on journeys both within and to/from Germany.

Seat reservations for daytime in both 1st and 2nd class can also be booked separately from tickets.

Happy Rail is a Netherlands rail ticket agency which sells tickets for both national and international journeys within a range of countries including Belgium France, Germany, Sweden, Switzerland.

HappyRail doesn't charge booking fees in the conventional sense, but if you're not Dutch you can expect to pay additional transaction fees, which will be added to the total cost.
That's because the most common form of online payment used in The Netherlands is 'iDEAL' and HappyRail doesn't charge a transaction fee for 'iDEAL' payments, but only Dutch residents can sign up to 'iDEAL'.
Other forms of payment include Visa debit cards will incur a fee.