Travelling with the application receipt for a French residence permit?

As far as France is concerned you can leave and re-enter the country (and the Schengen area) with a récépissé for a renewal (récépissé de renouvellement de carte de séjour) but not with a récépissé for a first application (récépissé de première demande de carte de séjour) nor with the document you get if you applied for asylum (which can be either a récépissé or an autorisation provisoire de séjour).

I am not sure to what extent Schengen border guards in other countries are aware of the details of these rules (or even if they are supposed to follow them at all) so if you transit through another Schengen country, you might have some trouble getting them (or an airline!) to accept the récépissé instead of the more familiar carte de séjour.

Except if you applied for asylum, I think you should also be allowed to travel in other Schengen countries but the same issue (familiarity with the document) might come up and it would seem prudent to wait for the actual carte de séjour if that's possible.

Conversely, if you don't need a visa, I would guess that you should be able to travel without relying on the récépissé. Similarly, if you are a Swiss or EU citizen, you could also receive a récépissé when applying for a carte de séjour (which is not mandatory in this case) but you can easily leave and come back with your passport and the récépissé does not really matter for you.

All these documents are also described in the Prado database usually with a short note on applicable rules.

Theoretically, you could also just leave even if your récépissé is not valid for travel and then apply for a new visa from a French consulate abroad but it takes time and if they refuse you would be in a very unpleasant situation, unable to re-enter France and possibly forfeiting your application. In this situation, you could still appeal a refusal but it can take two years and more so it's not really helpful in practice, even if you have a good case.

Finally, the webpage mentioned above describes another procedure I didn't know about for emergency cases: applying for a visa préfectoral before leaving. But that's only for exceptional cases and presumably not relevant for you.