I have not actually seen the race, though I have ridden many of the mountain stages in the Alps. A few random thoughts. I think there are really three options to see the race in person: On the side of the road during a mountain stage, in a town during a time trial, or in Paris for the final sprints near the Arc de Triomphe (someone check my spelling).
On a mountain climb the riders will go by individually, and slowly. Better for viewing. But the roads close for many hours; you are there for a long day. Bring a picnic or book a room at a mountaintop hotel on the route (if you can still find one). It's cool in the mountains, which could be a big advantage.
To see a timetrial you can stay in a town a 1 hour drive away - that's likely the closest you'll be able to find lodging. Drive over for the day and back later. Probably a long walk into town from where you'll find parking.
Paris is a nobrainer if you want to see the final stage and take in some of the sights. Spouses tend to like this option the best, but you'll have to go to the Louvre as payback.
The race covers a lot of ground, so it's actually a bit of work to keep up with it - well over 100miles most days. I've never taken a bicycle tour to see the race because it sounds too much like work - up at 5:30AM to get to the course, ride a few hours, watch the race go by, then shuttle in the van for a few hours to get near the next stage. Not enough time for beer drinking.
What is your wife's attitude toward long days on the side of the road spent with hordes of intoxicated European partiers? Could be the best thing ever, or the wurst,
depending on what you like. BTW, there's a great book "Motorcycle Journeys through the Alps and Beyond" that gives lots of helpful information on the roads, riding, and seeing Europe in general. Might be useful if you're serious.
- JimY