55km away from Nice is San Remo. An other famous race, Milano - San Remo (almost 300km) ends there. It happens in March, one week after Paris Nice. In 2015, it'll be back on its historical finish line, Via Roma.
The Tour de France came in Nice for the last time in 2013.
In 2015, the closest it'll get to Nice is on wednesday the 22th of July, in the 17th stage, Digne - Pra Loup. The closest point of the itinerary from Nice is Annot, 1H15 from Nice. This could the the opportunity to discover the scenic back country to see the Race. (Along the Castillon lake or in the slopes of the Allos pass, a little farther than Annot, for exemple.) Landscapes are wonderful but expect traffic jams to go back to the coast. Also plan to get there early enough, before the road is closed for the race (several hours before the race for the Tour de France). You can also use the Nice Digne little train to go to Annot. Details in Get to Nice.