The earliest match I could find in Google Books searches for get-go, git-go, getgo, and gitgo is one in Louise Moore, "Black Men vs. Black Women," in Liberator (August 1966) [combined snippets]: I want to try to explain how we Black women got into this bind.
Get-go and go-get are different idioms. From the get-go means 'from the very beginning', so get-go means 'beginning'. Go get, on the other hand, means to take action (go) and achieve (get) something in return. It's a serial verb construction, Go get me another brewski, would you? A go-getter is an old term for what is now called a self-motivating overachiever.
