Learn Russian: Мы не работаем, а отдыхаем.

They are both first conjugation verbs that end in -ать in their dictionary form (работать and отдыхать). Because the subject is мы (we), you drop the -ть and add the plural ending -ем to get работаем and отдыхаем.

Does Russian have a specific verb form for 'are working' versus 'work'?

No, Russian doesn't have a separate 'continuous' tense like English does. A present tense verb like работаем can mean either 'we work' (in general) or 'we are working' (right now). You just use the context of the sentence to figure out which one makes sense!

Why is а used for 'but' instead of но?

In Russian, а is used to show a strong contrast, especially when correcting a negative statement (not X, but rather Y). Since we are saying we are not working, but instead resting, а is the perfect fit. Но is used when adding a complication or conflicting idea (like 'I am resting, but I am still tired').

Do I need to repeat мы before отдыхаем?

No, you don't need to repeat the pronoun here. Just like in English, the мы at the beginning applies to both verbs in the sentence. Plus, the -ем ending on отдыхаем makes it completely clear that 'we' are the ones resting.