I got a job offer for a better paying job! Now I have to give my two weeks notice and I'm unsure how to properly go about this?

I had a job interview today and was given a job offer shortly afterwards for more than what I was asking for and its also quite a bit more per hour than what I am currently receiving. So I've decided to take the offer. Now I need to give my notice to my current job. I work in a fairly small office, family owned with less than 20 people in the office. Our HR gal is going to be out of the office tomorrow and she is who I typically thought I would give my notice too. I would like to give it tomorrow so that way my last day can be on a Friday in two weeks. My supervisor will be in but our office is set up like cubicles so there is no where to really have privacy I've been so severely underpaid and have watched new people come in that I've had to train that have been paid significantly higher than I. I'm glad to have finally found a way to remove myself from this situation after the last 2.5 years. I will miss my coworkers DEARLY, I'll say that for sure.

I haven't done this in forever and I'm so nervous! What should my resignation letter say? Can I wait until Monday to give it to HR or should I just give it to my supervisor in front of everyone? I would prefer privacy. I need help with this because I am starting to majorly overthink this entire situation.

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