When I want choose a date for, say, a party, does it sound natural to say the following?
Let's fix a schedule for the party.
If that sounds ok, then could I change fix for set?
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A schedule is rather more than just fixing a date. It's more appropriate for a series of events, such as a conference with registration, talks, meals etc.
If you are choosing a date for some event, that's not a schedule, its just, well, a date.
Let's fix a date for the party.
Although more idiomatic (for me) would be
Let's set a date for the party.