I would tell you manager and deliver your letter at the same time, with the last date you want to work on there. It’s the most professional and polite way to go about it.

People at my work have just been writing letters and leaving them in the back with their names

SM or ASM canned response will be that they need it in writing, with it dated and stating when your last day is (keep a copy for yourself). Might as well present your SM with it in writing when you tell them. This is just a courtesy, and my advice is to avoid giving any additional information about why or where you’re going. If it’s true, maybe thank them for the opportunity to work there and what you were able to learn. Again, no reason to add any flowery language that is actually the opposite of how you feel.

SM has every right to let you go immediately, as you have every right to leave before the intended date if there is any sort of retaliation or bad blood. These are courtesies extended to those deserving. Use your best judgement and good luck with your future endeavors. We support you!

Generally a written note with your end date is normal with any place of employment. If you're ever uneasy about it, you can take a picture of the note on your boss' desk, send that picture to them, and keep a screenshot of that message.

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In person then follow up by email.