How to take the rail journey from Lyon to Frankfurt (Main)

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.

On journeys within Germany and on most international journeys from and to Germany on trains operated by DB:

• For journeys by the express trains, EC. IC and ICE trains an adult (on German railways an adult is a person aged 15 and over) can take up to four children aged 6 - 14 with them at no additional charge.
• Tickets need to be booked for children aged 0 - 5, but there is no charge for them.
• Those aged 65 and over can take journeys at a discounted rate.
• Dogs, which are too large to be placed in pet carriers, require tickets which cost 50% of the adult rate.

When booking both types of the discounted 'Sparpreis' tickets seat reservations are an optional extra, so need to be added when booking if you want an assigned seat(s).

SNCF is the national rail operator in France and not so long ago it recently simplified both the range of tickets and the term and conditions of using them.
Therefore the cheapest tickets also become more flexible too.

Other improvements included a simple integration of travelling with bicycles and rail pass users being able to book reservations for national journeys, without paying booking fees.

And it also recently launched SNCF Connect as replacement for its Oui.SNCF booking service.
You can set up a low price alert to let you know when the cheapest price is available for a future journey.

Something else to look out for is that simplest way of travelling with tickets is to use the SNCF Connect App, as booked tickets can therefore be stored on your mobile device, which can then be shown to the train conductors as proof of purchase.

SNCF Connect on Google Play

SNCF Connect in the App (Apple) store

SNCF also operates RailEurope, which is in effect a dedicated online booking service for making bookings from outside of Europe.
If you reside outside of Europe you may discover that SNCF Connect rejects phone numbers or card numbers, but RailEurope won't do so.
Though RailEurope will add booking fees when making a purchase

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.