The thing to know is that the itinerary that must be provided to obtain the visa is never verified after the fact, so no need to worry about that. A real plus before making this trip is to learn the Cyrillic alphabet: it is very useful when traveling to Russia. Train signs in stations are now much translated into the Latin alphabet, but this is not yet the case everywhere.
Then you have to book your plane tickets. For my part, I always do a comparison on most search engines like Skyscanner, Liligo, Opodo or Bourse des Vols. You should know that you can easily enter Russia with a one-way ticket without having an exit ticket.
Independent or organized trip? It is quite possible to organize your trip to Russia yourself, but for those who prefer to go through an agency, you can request a free quote from Evaneos, which offers individual à la carte packages. Otherwise I would also recommend Russia Autrement, a small French-speaking agency specializing in Russia.
Screenshot of Russian Railways website: you can choose the berth you want
Buying train tickets for a Trans-Siberian journey is much easier now than before as it can be done directly on the Russian Railways English site. The site is pretty well done and you just have to print your electronic tickets or download them to your smartphone to board the trains. Train times are now in local times instead of Moscow times as they were before 2018. One of the niceties is that city names are sometimes spelled differently. For Moscow or Vladivostok it is not too complicated because it gives Moskva or Vladivostok, but there are a few trap towns. To find Khabarovsk for example, you will have to enter Habarovsk, for Yekaterinburg, you will have to put Ekaterinburg or for Ulan-Ude, Ulan-Ude. Tickets can be purchased up to 60 days in advance. Except for those who speak Russian, I would advise against trying to buy them at the ticket counters at the stations as no one speaks English. In addition, on the site, you can easily see the days, times, prices and availability, so it allows you to organize your trip quietly in front of your computer or smartphone. Once you have chosen your place and entered your data, you just have to uncheck the insurance boxes and you can pay by credit card.
3rd class berths on the Trans-Siberian: there is room at the bottom, but very little at the top …
The trickiest thing then is to choose your bunk. There are three classes in Russian trains: first class, which allows for a compartment with two berths; second class (or coupe) with a four-berth compartment and third class (platzkart or open sleeping) which is a dormitory car in a way. To get an idea of the budget, for a Moscow-Vladivostok, the 1st class is from 40.