Learn French: Nous recevons des médailles et des diplômes.

The infinitive is recevoir (to receive). In forms like je reçois (I receive), the cedilla is needed to keep the soft 's' sound before the vowel o. In nous recevons, the c is followed by an e, which naturally makes a soft 's' sound, so no cedilla is needed.

Why do we need to put des in front of both médailles and diplômes?

In French, articles like des (some/plural of 'a') must be repeated before every noun in a list. While in English we can group them and just say 'medals and diplomas', French requires des médailles et des diplômes (literally 'some medals and some diplomas').

Are médailles and diplômes masculine or feminine?

médaille is feminine (une médaille), and diplôme is masculine (un diplôme). When they are plural, like in this sentence, they both use the plural article des, so the gender isn't obvious just from looking at the article.

What does the little hat accent (circumflex) on diplômes do?