What ACTUALLY happens on Oct 1? I have heard a number of things. How about if I buy something from a neighbor? What will I need to make me legal? Neighbor has a nice 9MM that I am thinking about.
You can't take possession of most handguns on or after October 1st unless you've gotten a handgun qualification license, even if you're getting it from a neighbor (or other private seller) and even if you purchased the handgun or began the background check process before October 1st. So, you need to begin that process soon enough for 7 days to have passed before October 1st. You also need to make sure the seller will give you the handgun on the 8th day even if the application hasn't been returned by the MSP as not disapproved (they never actually approve you), which the seller can do.
If you don't already own a regulated firearm, getting a handgun qualification license may require you to, among other things, go through training. Even if you do own a regulated firearm already, you'll have to pay a fee and submit fingerprints to get the license.
For the firearms that aren't allowed to be sold after October 1st (e.g. AR-15s), you can take possession of them later so long as you began the process prior to that date. It's only the handguns for which you need to take possession before October 1st.