In France, there are currently 4 operators of intercity trains: SNCF, Eurostar, Thalys and Trenitalia. Of these, all except Trenitalia accept Interrail (on most trains).
SNCF is the state railway in France, they operate everything from high speed to suburban trains.
Assistance for disabled travellers: https://www.sncf.com/en/passenger-offer/travel-for-everyone/accessibility
High speed TGV trains come in two variants, the more common TGV InOui, and the low cost TGV OuiGo. Interrail is only valid on TGV InOui, with mandatory reservations that cost around 10-35€ (20 or more for most international routes). On domestic TGV InOui, the cost rises from 10 to 20€ as more seats get sold on a particular train. OuiGo tickets are very cheap, it can sometimes be less than the InOui reservation (you can buy them here). Platforms that TGV trains depart from have controlled access, you can only get to the platform with a valid ticket and reservation. SImilar conditions as for TGV InOui apply to TGV Lyria trains that connect France and Switzerland, and German ICE trains that connect France and Germany.
Sometimes, you can avoid the high cost of international reservations by taking a TGV to the last French station (Strasbourg, Mulhouse, Lille…) and continuing by regional train, or making the entire trip by regional trains, if you don’t mind a slower trip.
You can get the reservations online at this Belgian railways website (domestic TGV InOui only), or through the Interrail website/app (all TGV InOui including international).
You can also obtain reservations for all SNCF trains by phoning SNCF – phone number and opening hours are at https://www.sncf.com/en/customer-service/contact-us/telephone. If you press “#85” after the automated voice begins this will change the language to English. Reservations can be emailed to you for immediate usage.
Ticket offices in France and in some other countries such as Germany or Switzerland also offer SNCF reservations.
Intercités are low speed intercity trains, they also require reservations, which cost 10€.
Intercités de nuit & Nightjet
Intercités de nuit are sleeper trains operated by SNCF, which run from Paris to southern France. There is more information in this wiki section.
An ÖBB Nightjet train also runs from Paris to Vienna, though only 3 days a week. There is more information in this wiki section.
Transport express régional (TER)
These are regional trains in France, you can use them with Interrail without a reservation, with a few exceptions. TER from Paris to Normandy do require reservations.
Transport express régional – Grande Vitesse (TER-GV)
Some trains in the North East of France are classified as TER-GV. These are parts of TGV trains which stop at a few more stations acting more like TER trains. Interrail is valid on these services on the payment of a €2 supplement which can be purchased through: https://www.ter.sncf.com/hauts-de-france/tarifs-et-cartes/billets/option-grande-vitesse
Réseau express régional (RER) and Transilien
These are suburban trains around Paris, Interrail is not valid on these.
Eurostar and Thalys
These operators run high speed international trains to the UK, Belgium, the Netherlands and Germany. Similar to other high speed trains in France, you they only accept Interrail with a mandatory reservation, in this case it costs around 25 or 30€. There is some more information on Eurostar and other options between the UK and France here: Eurostar
Trenitalia runs a few Frecciarossa high speed trains from Paris to Italy, but they don’t accept Interrail at all on this service. International TGV trains to Italy that run in cooperation between SNCF and Trenitalia do accept Interrail, but if sometimes the reservation for those can be similarly priced to a regular ticket for the Frecciarossa train (if you buy it a long time before departure). You can check the price at https://www.trenitalia.com/.