If you are talking about a standard Western calendar that you buy in a store, then I would hazard that カレンダー is safe to use. If you want to talk about other calendar systems, such as lunar, then use 暦.
For asking about a date, you say 何日, as in:
今日は何日ですか？ What is today's date?
For asking about what day, you use 何曜日, as in:
来週の何曜日がお暇ですか？ What day are you free next week?
For a business setting, one can use 日付:
Please let me know if there is a change of date.
And for a legal or official setting one can use 期日 for fixed date/settlement date:
期日の変更の申立ては、期日の変更を必要とする事由を明らかにしてしなければならない。 A petition for a change of date shall be filed by clarifying the grounds for requiring the change of date.