I'm just having different ideas about the logo size. Some places say it must be 800×800 px and 72 dpi.
What's the artboard size I choose first? What are resolutions I should use? What is the best DPI amount I should use?
Start building the logo in a vector application, and worry about artboard size and resolution later: with a vector format logo, your artwork will be scalable to ANY artboard size and ANY resolution.
Don't know what a vector application is? Here's a list of the more popular ones:
Is Photoshop a vector application? No.
What should have in a Scalable logo?
I will answer you what it should not have.
It must be 800×800 px
It should not have a fixed pixel dimension.
and 72 dpi.
It should not have a specific PPI resolution.
If you use a vector-based program as Lucian commented, you could start with the default artboard, A4, letter for example.
After finishing the logo, now you export it to a raster image, let us say to 800px width or whatever size you need it.
But that is a different step, the delivery.
Some places say it must be
This sounds like you are a very specific usage, for example, a "logo for xxxxx template on WordPress" but you do not design a logo based on a specific case, you design a logo to be used in different places.
and 72 dpi.
A web page does not use dpi, if you need 800px you need 800px regardless of the PPI.