I want to prove a theorem using the result of a well-known theorem. Should I write the well-known theorem as a lemma since it aids the proof of my theorem or as a theorem?
It depends on how well-known it is. If it's genuinely well-known, then I would just say "by Theorem XYZ," then the implication of the theorem as it is relevant to your proof, with any citations for the theorem as necessary. The less well-known it is, at least insofar as your audience is concerned, the more you should probably include, whether it be the statement of the theorem, or even the theorem and its proof as well (assuming it's not a hundred pages long or something).
As for what to call it - generally if it's referred to as a theorem, I'd just call it a theorem to avoid complications.
In short, though, it simply depends and you'll have to use your best judgment. I doubt you could really go wrong by stating the theorem at least, for clarity's sake if nothing else, but for really well-known theorems (e.g. Fermat's Last Theorem) that wouldn't even be necessary for the average mathematically-inclined person.