At the risk of sounding ridiculous ..... have you checked the fluid level.
I too bought a car, got the engine running, took it out for it's first drive, it slipped a lot.
Checked the fluid, it was low. For the record, I tried to check the fluid before I drove it for the first time, but the car needs to be at operating temperature to get an accurate reading.
Filled it up with synthetic RedLine D4 ATF (it took a little over a quart), drove it a while, and now it shifts like the best auto I have ever driven.
Good luck, whatever it is, you'll love it once you sort it out.
I apologize for making such a silly suggestion, but there are two ideas here. One is just the low fluid. The other is the synthetic RedLine. It might have freed up some components once it circulated around in the trans. (I do not work for RedLine, or profit in any way from the sale of RedLine products.)
Mark