I may have asked this before, but as is always the case with me, things don't stick. This is how I've written the sentence. In fact it's always how I write sentences of this kind, but I'm sure it's not correct. Question is, where on earth does the question mark come, and how does it affect capitalisation? Problem here is 'to' (following the question mark) should be capitalised, strictly speaking, but then it looks and reads awkward. I suspect there's a word for when additional information is inserted between dashes, but I don't know it.
If I wrote it, I would change the phrasing so it wouldn't be a question. If it was staying as you've written it, I'd put another dash after the question mark to separate it, but I have no idea if that's correct (probably not).
That certainly looks cleaner, and as simple an answer you may think it is, suspect this may be correct.
If it is, I'm genuinely shocked. Happy to help though Edit: it looks good to me though, so if it was me and I couldn't rephrase, that is what I would do.
I went to the shops—or should that be store given I'm in America?—to buy some groceries. or (as mentioned) I went to the shops (or should that be store given I'm in America?) to buy some groceries. That type of dash is called a parenthetical em dash. It basically is parentheses. So you choose whichever fits your style. I'd italicize 'store' though.
Incorrect usage! Commit yourself to crappy - it was just a crappy example. Or don't - it was just an example. The in-between route is an affront to grammar.