When I get ready to create the logo, do I have to change the the pixels of the group icon? Would it be bad quality?
Kinda confused because I had a 1920 x 1080 design and I tried to resize to a group icon but it looked terrible; 
So it would be best to change the pixels to the group icon when i first begin the logo?
Yes, it is best to change the pixels then make the logo. If you don’t then it will just looked squished every time.
For icons to fit perfectly it needs to have be a square, the resolution would be: 1080 x 1080 or any resolution that has the same pixels vertically and horizontally. Roblox stretches your image to look square that’s why it looks like that.
All group icons are prefect squares. Simply crop your image’s sides until it is a square, then try uploading it.