The auto in the GR86 is quite an old style one, that essentially uses liquid to engage the gears (imagine a water wheel - kind of), what that means is that there is always a bit of a lag because the transfer of liquid and build up of torque is slow.
There is a great variation in how good/bad these type of autos are. The GR86 has a really good one, so its not to 'slushy' or 'laggy', and the electronics are really quite good at doing the right thing on the track - which is NOT easy for a car with comparable (to sports cars) low power. High performance cars might have worse electronics but you wouldn't know because they just have so much power in all the gears.
DSG and the other things are essentially like robotics made to change gears using a clutch - they can generally always beat a human because they cheat, and are not limited to working with just one clutch
Neither as as quick and not _quite_ the brutal engagement of a dog box on a sequential manual though - they will always be faster, but they are really track and racing only.
Auto or not, this is the ultimate drivers car (within reason) - your still rear wheel drive and still running on this magnificent low and tight platform.
One thing about auto's though - dont stick to the book for changing your automatics transmission oil. Unlike a manual, they a FUCKING EXPENSIVE to fix, and while transmission oil isn't cheap, do not hesitate to get it replaced at half the interval it says for you to to. These older autos will last forever if you just keep replacing the oil fairly regularly.