What is your departure time in Montélimar?

If the bus has the line number X71, it has SNCF (regional train) tariff (10.20 / 15.20 / 20.20), if the bus has the number D36 on it (on the same route), then you’ll have to pay 5,50 € as an adult (6.30 / 8.45 / 17.15 / 18.20).

At least the german version of the app does call the bus a bus. Unfortunately, you can’t be sure that it’s free to use just because it’s shown by the app - there are a lot of mistakes. 

Thank you!

How can you find the line number of the bus? In the app all information I can see is “BUS59585”. 

Are regional (TER) trains included in the interrail ticket? 

How does one in the best way find information about the trains, the platforms and eventual changes? I understand that the most reliable source is the train company itself, but how does one easily find which train company drives what train? 

Thank you 

Departure time from Montelimar is 15:20 the 23 of January. 

Unfortunately, the app isn’t the best way to plan a journey. Not even the second best way… *g*

Here, I just did know to which region of France you are planning to go and did know where to have a look. 

Yes, they are. TER trains are the regional trains of french railway company SNCF. For participating railway companies, you may get an overview here: Railway Companies | Interrail.eu

There are a lot of better travel planner to get specific information. For instance, austrian railways (ÖBB) travel planner does show which company operates a train or bus. In your case (last line; Betreiber means operator):

Hej Debora!

Här är lite tips om du inte redan har sett dem. 

Here is some useful information from the experienced travellers in the Community regarding both planning, reservations and activation of pass and travel days. 

 Planning

The rail planner is normally not up to date, as it only is updated once a month, so to be sure of the time table you better check the timetable and availability on the websites of the national railways. The bigger national railways, like DB (Germany) SBB (Switzerland) and ÖBB (Austria) cover several countries. 

 Reservations 

France is a bit confusing. A lot of smaller railway lines can shut down for months, if not years for renovation works. Or it just closes down for rail traffic because the line is chronically underfunded, and not safe for trains any more. Rail replacement buses are usually in place, like in this situation.

SNCF is the operator, so your interrail should be valid on it. If the bus is shown as a TER or BUS (+ 5-6 numbers), it's should always a rail replacement bus operated by SNCF. No need for extra tickets.

It's a bit difficult to find your way in French railway planners, it's a bit confusing since it's very regionally arranged. Not always customer friendly... Perhaps the website of TER PACA (Provence-Alpes-Côte d'Azur) or SNCF Connect has a proper planner for your travel wishes.

In the past TER-BUSes were also covered by the pass-they ran as train-replacing. But the last yrs the regions have mostly taken over and reorganised them into the already existing other regional bus network-often with very low fares, and stopped taking passes. This thus now varies per region and it may even turn out as Hektor explains-that the 2 run side by side. But anyway-the expense tends to be quite low so not to be an enormous hurdle.

Thank you so much for fast and informative answers! 

Would you suggest I do my research very well concerning what operators run the trains and see if I can follow as good as possible where I might have to pay extra. Or is it more advised to just go and solve things as they come? 

If you are not on a tight budget just go and see what comes. First try to show your interrail pass and if it I not valid you can still buy a normal ticket. 
but if you just use 2-3 buses you can easily check on oebb website as suggested by some other user. If SNCF is operating a Bus as train replacement then mostly interrail will be valid.

I visited the busstop in Valreas last year, that’s a town on the line to Nyons somewhere between Montelimar en Nyons. You can see on the screenshot it is  a SNCF-bus as replacement of a train. In combination with the info on the interrailapp. it is nearly sure that interrail will be accepted. “Trainreplacement bus” (in French “car”), the last train Pierrelatte-Valreas-Nyons ran in 1951………………….!!

And if the driver looks puzzled, just say it's an SNCF ticket. Works for me every time.