Stamp can have different meaning in French, so without context it is difficult to give a single answer. So here are the most common:

postage stamp = timbre poste (the piece of paper that you stick on a post card). Then you can use the verb "affranchir" (meaning that you sticked enough stamp to pay the post) or more simply, "timbrer".

Stamping device or printed mark = cachet, tampon (in french, they both mean the tool and the mark of the tool). Then you can use the verbs "tamponner" or "cacheter", e.g "tamponner un passeport". For an official stamp by the post (e.g. again, on top of the postage stamp) you can use "oblitérer". For the stamping of public transportation tickets in countries where it is necessary, you can use "composter", e.g. "J’ai déjà composté mon titre de transport".