What punctuation should I use here?
Hester needs to be more grateful and focus on: her baby, Pearl; and her sewing.
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Because this is ambiguous in writing — albeit not in speech due to intonation:
You should therefore resolve your Oxford-comma conundrum this way:
Notice how I’ve repeated your preposition instead of risking confusion in how it distributes?
The moral of our story is that one tiny word, deftly added, is better at clearing up theoretical confusion than silent punctuation is.
A colon should be used when you're signaling that you're about to elaborate on something or about to write a long list of things. For example, this sentence needs a colon:
Hester needs to focus on the following: her baby Pearl and her sewing.
In your sentence, however, "her baby" is the object of the verb "to focus", so there should be no colon between them.
Semicolons are used to signal a longer pause in a sentence, or between two separate sentences if the writer feels like combining them into one for stylistic reasons. For example, this sentence needs a semicolon:
Hester needs to be more grateful; she needs to focus on her baby Pearl and her sewing.
It depends on whether "her baby" and "Pearl" are the same person or not. If they are the same person, skipping the comma would be best for the sake of clarity:
Hester needs to be more grateful and focus on her baby Pearl and her sewing.
It could be argued that "Pearl" should be offset by commas because it is a parenthetical element. However, I think that this is not strictly necessary in this case, and the sentence is more clear without the commas.
What if they are two separate people? (Not likely if this is about The Scarlet Letter, but I'll explain just for the sake of being thorough.) In that case it becomes a list of 3 things, and the commas are required. And to ensure that there is no confusion, I would also move the items around to make it clear that Pearl and the baby are two different people:
Hester needs to be more grateful and focus on her sewing, her baby, and Pearl.