Here are my 2 cents:
Stage 4 Col Springs Park car near Uintah/Mesa/19th, easy parking, mix of residential and retail. Ride bike into Garden of Gods. This is a must see area because of the beauty. After (say) 2 laps of the circuit, ride bike to downtown Col Springs and the finish area. In 2011, they had valet bike parking near the finish. After race, plenty of food and drink in downtown, then ride bike back to car.
Stage 5 Woodland Park to Breck I suggest staying in (say) Vail, Avon or Copper and riding your bike along the trails to Breck. The bike trail is wonderful and save the parking hassle for 2 days. The bike trail in Breck is parallel to the route, so the road will be close but you can easily bike side street and the bike trail to get around.
Stage 6 Vail TT See stage 5 note, stay in Vail or Avon to make it easy. In the AM, you can bike the TT course and/or surrounding areas They probably close the TT course 2-3 hours in advance of the race, so bring food, booze, water, etc. and hang out with the crowds along the route.
Stage 7 Finish in Denver Urban finish with large "festival" type atmosphere. No need to arrive 3 hours early, unless you MUST have spot at finish line. Most of the finish line is filled with VIP stuff. Not sure there are any free seats at the finish line, so likely you will 25-50 yards away. I have shown up 30-45 minutes before finish and stood 3-4-5 deep in crowds and had a ton of fun.

Hope this helps.

That's GREAT information. Thank you!

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-Stephen in Arkansas

We went to the Vail TT last year and it was a blast. We showed up 1-2 hours before the race started to beat the road closures. We had plenty of time to wander the event village and the team camp. The busses, cars, etc are all right out in the open and you can more or less walk up to the door and shake the hand of whomever is out there (Jens Voigt is a champ with the crowds and nearly missed his start to give an autograph for me). Froome, Sagan, Tejay, etc all at an arm's reach. Pretty cool. We drove over from Denver (long day) day of and came back that night.

We never made it to a road stage but for me the TT has the greatest value if you have to pick one stage. It gives you maximum exposure to the athlese and any of the cool swag you want to see while keeping you close to civilization. And Vail is just awesome in general.

Very cool. For the TT stage, and if we're staying Vail like was suggested by bobincolorado, would you mess w/ a bike that day? Or just walk everywhere?

Trying to weigh renting for a couple days vs flying our bikes out there.

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-Stephen in Arkansas

From parking to the start line, restaurants, team camp, etc is all very walkable and so packed with people a bike wouldn't be a ton of use. That said, if you want to check out the course or camp out high up on the pass to watch the race you will need your bike. Not riding the course was one of my regrets.

That's all I need to know.

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-Stephen in Arkansas

In my opinion....and from personal experience....the best way is to have a bike. Riding in Colorado, does it get any better?

Col Springs. Yes, a car is fine, but with bike you can see 2 of the race locations, and much less parking/walking hassle.

Breck. The bike path from Vail to Breck is one of the best in the country...try it you will like it. I have never watched the race finish by driving to Breck and imagine (but I could be wrong) parking is challenging for the day of the race.

Vail. The other post is accurate, the start in the village is walkable and enjoyable. Why not ride the TT course at 7 or 8am? Then, check out the village with team buses, riders on trainers, start gate, etc. and ride (depends on exact road closure times) back up the TT course to see the guys in action. Timing here can be tight. The "main" uphill portion (above Gore Creek Campground) of the TT course is 7-8 miles away. Walking will take a long time versus bike. Worse case scenario, you ride the course in the AM and enjoy everything in the village in the afternoon.