Trying to find out if phrases like "went and got" are correct, e.g.:
She went and got the book.
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Absolutely. "Went and got" match and can therefore be used together. As a simple test, you can try "she went," and also "she got," so putting them together is fine. If you said: "She went to the store and got the book," you probably wouldn't even question the phrase. Dropping "to the store" is fine.
I think in American English, OP's example probably comes across as slightly odd, bordering on tautologous. But certainly lots of Brits have no problem with, for example, He went and left me, particularly in an informal spoken context.
That's because in British English, we often use to go and [do it] meaning to proceed [to do it] (often, in contexts where doing it was unexpected, amusing, and/or offensive to the speaker). But I don't think the usage in common in AmE.
My advice to OP would be to avoid the doubled verb form unless it's important to convey the sense of both leaving and returning/fetching. That same advice is in the example from The Dimwit's Dictionary which I linked to above, where Robert Hartwell Fiske (endorsed by William Safire) says "DELETE went and".
It sounds a bit crude when you could have used 'went to get' but it's equally valid English.
They do have different connotations; "went and got" implies the description of an action in more of a narrative context, following the action step by step (and possibly in the first person); "went to get" is more clearly past tense and - at least to me - infers a summarisation of a potentially complex set of actions.
It's all open to interpretation though!