This is normal. Try shift at a slightly higher RPM, Im betting youre shifting at 3.5K or under. Until you get used to the car, it will "click" The car has 9K of revs so it likes shifting at the higher RPMS. If it shifts fine at 4-5K, try slowing down the 1-2 shift.

go 1-neutral-pause-2 instead of 1-neutral-2


as for the whining noise, I'd also bet its a differential. Just drive it til it blows and pick up a new one while you're waiting...

 

We bought my daughter's 2001 with 95k miles. After another 5k or so the tranny started to whine. After a great deal of diagnosis by the dealer and by a mechanic that I trust (sounded like maybe diff, maybe tranny, maybe clutch pilot, maybe...?), we concluded that the dumbass previous owner had put seriously wrong fluid in it when changed and all of the bearings were toast. He probably saved a few bucks by not using Honda MTF and only cost me about a grand.

Just goes to show - never assume that correct maintenance has been done; change all fluids when buying a used S2000.