The Bridgestone RE-11s (Extreme performance summer) and S-04's (max performance summer) both rock according to kenchan. Tire Rack agrees. I have no experience with either but will roll on kenchan's advice. However, neither are recommended for winter/snow. So if you can afford a summer set of tires and then a winter set as well (with wheels) go for that. Otherwise buy something more attuned to all season, recognizing that it will impact overall handling if you do. Tirerack.com is your friend.
Wheels & Tires - Nissan 370Z Forum
As for the handling charactereistics of the sport versus base model... I have never driven a base model. I can say that the sport model does give significant feedback when it goes over bridge joints and cracks. Tire pressure, as DR pointed out, might help but for day to day driving, I run standard tire pressures. Recognize that the suspension, wheels, and tires on the sport are more firm/agressive than the base model. This may be part of the reason you felt what was going on. I love that aspect. Feedback is great. Here is a thread with base/sport handling differences. You'll have to weed out the crap posts.
Base vs. Sport Package
Of course tires are a huge part. Here is an interesting...
... thread about tires that ARE crap!
Are these tires any good? Hercules Raptis Wr1's (if you have to ask ... NO!)
I'd say buy it. Cuz, as someone once said, race car...
