Learn Hardcore Russian: После приёма я буду полоскать рот тёплой водой. - After the appointment I will rinse my mouth with warm water.

После requires the genitive case (родительный падеж) when it means after.
So приём (nominative) → приёма (genitive): после приёма = after the intake/appointment/reception.

What does приём mean here? Is it “appointment” or “taking medicine”?

Приём is a very common noun with several meanings, and context decides:

приём у врача = a doctor’s appointment/consultation
приём пищи = a meal
приём лекарства = taking medicine (a dose/intake)
приём гостей = receiving guests

In your sentence, После приёма is most naturally understood as after taking (something), often medicine, unless the broader context says it’s a doctor’s appointment, etc. If you want to remove ambiguity, you can specify:

После приёма лекарства…
После приёма у врача…
После приёма пищи…

Why is the future formed as я буду полоскать instead of a single future verb?