1. Our first visit will be June 23, next Saturday.
2. Our first visit will be on June 23, next Saturday.
3. Our first visit will be on June 23, on next Saturday.
4. Our first visit will be on June 23 on next Saturday.
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Which ones are grammatical and commonly used?
2 is best. 1 is okay, too.
3 and 4 have repetitions in them ("on") and wouldn't be used in normal conversations.