I would do the suspension first, including being able to dial in camber, and get race tires and lightweight wheels. Also at the same time while the suspension is going on I would put on new brake pads, SS lines, new brake fluid (this is easy).
Then I would actually track the car until I felt that the handling was perfect. You may want to, in the mean time, put in race seats and 4 point harnesses inside. Personally I would put in a race steering wheel with no airbag, some pedal covers too.
If you want to do weight reduction this is up to you but I wouldn't destroy the car too much.
LAST thing I would do after running the car maybe for a while would be the 3.8 swap and full exhaust, all at once.
If it was me though I would not buy a Cayman as a track car. I would buy a Lotus or an open wheel car or toy car like a Caterham CSR
