For example, if I was saying something like “Did you buy the groceries?” and then I wanted to add “Or do you want me to?” how would I translate the “or” at the beginning?
Another example: “Do you like my new car? Or did you not see it?”
My only understanding of “or” in Japanese is か but that definitely would not work here.
The closest equivalent would be それとも in both cases. "Or do you want me to (do it)?" could be translated as 「それとも私がやりましょうか？」, while "Or did you not see it?" could be expressed like 「それともまだ見て（い）ないんですか？」 etc.
While それとも is completely correct, さもないと or さもなければ is another expression you can use which is more formal. It is often given as "otherwise" or "or else," but of course there are many English sentences where "or" begins the sentence with those meanings. For instance, Weblio has the example sentence しっかりしろ, さもなければやめてしまえ (shape up or get out), which it attributes to Kenkyusha.
