Quick answer
Drinks take пить + accusative. A toast uses за + accusative: за здоро́вье, за роди́телей.
Soft drinks
Alcohol
At the table
The rules at the table
Three that matter. Toasts come before drinking, not after — somebody says a few words and everyone drinks together. There is always заку́ска, something to eat immediately after. And nobody pours for themselves if there is company: you fill your neighbour’s glass and they fill yours.
Refusing without awkwardness
Я за рулём (I’m driving) and Я принима́ю лека́рства (I’m on medication) are both accepted immediately and without comment. A vague я не пью may be met with persuasion; a concrete reason ends the conversation. You can also raise the glass, join the toast and not drink — that is perfectly acceptable.
Not “cheers”
Russian has no routine word for “cheers”. На здоро́вье is a reply to thanks for food, not a toast. What people actually say is a short toast — за встре́чу, за хозя́йку, за тех, кто не с на́ми — or simply Ну, дава́йте!.
Typical mistakes
На здоро́вье! — as a toast
За здоро́вье!
на здоро́вье is “you’re welcome” after a meal.
Я пью во́дку — as a general statement to a host
Я не пью / Я за рулём.
A clear reason is the polite way out.
Check yourself
Which is a toast?