Nice is in France and the Grand Prix is in Monaco/Monte Carlo - catch the #100 bus or catch a train there. Takes around 45 mins on the bus - nice views. For expensive cars go to Casino Square in Monte Carlo to see the affluent lifestyle.
There are various car/GP attractions in Monaco. You can walk around the GP circuit (free!); there is usually a fleet of convertible Ferraris based at Rascasse corner and at the swimming pool offering trips around the circuit, 45 euros as a passenger or 85 euros to drive (insurance included), take your licence. There are a number of shops offering GP souvenirs, T-shirts, etc, both up at the palace and at the exit of the tunnel. Authentic team merchandise (as opposed to Monaco GP merchandise) is also available but you have to look a bit harder and pay accordingly.
Perhaps the best little known gem is Prince Rainier's classic car collection in the Fontveille area; only a few euros to get in and a great collection of classic, mainly American, cars and including an F1 Ferrari from about 1990. When we were there it was virtually deserted and an oasis of air-conditioned coolness in the sumer heat. Look it up on the web, well worth a visit.