We think the Trainline ticket booking service makes it easy to make bike reservations online from outside France.
Firstly, note that not all trains require bike reservations but long-distance TGV services and some inter-city trains do. See here for an overview of which trains need reservations.
The results page includes a box for bicycle carriage if it is possible on your chosen journey. Note by 'bike carriage' I mean that you can take your bike FULLY ASSEMBLED onto these trains.
The search results may also include some bus journeys, so make sure you check the total journey times before agreeing to buy a ticket for a 23-hour journey from near Bordeaux to Paris like on the budget OuiBus result above.
Note there is also some variance in the cost of the bike compartment so it's worth comparing journeys.
From here it's a matter of proceeding to final booking and payment.
As you can see, you (usually) need to print tickets at a train station ticket machine. That means the system isn't fool-proof – if your departure station is small, it may not have a machine (usually limited to smaller regional TER stations like the one I've searched before).
If this is the case, you will need to visit a larger station or a ticket office. Alternatively, just buy a ticket online for the 'main' part of the journey (e.g TGV or Intercite train) and get the regional ticket at the station when you arrive (regional TER trains don't require bike reservations anyway - see here for guidance).
There is also no capacity to book tandems or other special bikes – for this information you will still need to visit a station, call SNCF (UK or France) or call Rail Europe if you are in the US.
All in all though, Trainline is a really great, easy to use option for reserving bike spaces on trains in France and also in other European countries.