it all depends on how much time you have and what part of the country you would like to visit. france is big and if you don't have much time to follow the tour you should pick towns that offer other interesting experiences close to where you want to go. the mountains are beautiful and offer alpine type activities, but there is not much else to do there on the cultural tip (mueseums, architecture, shopping, nightlife, etc.). if you have your bike and are fit the mountain stages are the most fun to ride and then hang out near the tops of the climbs to cheer on the riders. you won't be able to drive up to these spots unless you go the day before, which is kind of a waste of time if you are only there for a short time. and if you do you will need to camp since the hotels in any mountain finish town will be booked up by the teams and press long before the announcement of the route. i like the finish lines the best since they have the big screen so you can watch the race unfold before it comes into town and there are all the vendors and the press circus going on. and watching sprint finishes on flat stages, although over fast, are extremely exciting. since they close the course to bikes about an hour before the advertising caravan comes through riding the last part of the course will offer you the most time for riding. i like starting my ride at the finish and going backwards for 20 or so kilometers and then turn around and head back to the finish. time trials are also fun to watch since each rider goes by with a 3 minute gap between riders so you can pick out your favorite riders a lot easier. paris for the finish is very exciting, especially since they do laps around the most beautiful part of the city (louvre, champs elyses, siene river)....but it is very crowded.
if you have the funds renting a caravan type vehicle with satelite tv is a great option, then you can camp comfortably and watch the whole race from anywhere on the course. i believe germany is the best place to rent these type of vehicles due to laws, insurance and other costs.