Obviously, there is no need to give any tips on watching the race at either the stage starts of stage finishes. Watching the race on the road is a bit more difficult and there are some points to consider. Above all, you don't want to endanger the safety of the race nor do you want to cause a delay which may effect the outcome. The key here is to be respectful of the event and the riders.
Yes, it is a free country and you have every right to be there on the side of the road cheering your favorite riders, but please bear in mind that these are professional cyclists doing their job. You wouldn't want somebody coming to your place of work and causing you to perform badly so please don't create a situation on the road which causes the riders to perform their job badly.
That means staying well back of the riders as they pass. The pros tell me that every so-called "pat on the back" feels like a kidney punch. Just give them their space and let them ride their hearts out. It's OK to yell encouragement, take photos and even paint names on the road, but touching and getting in their face is a no-no.
If you are going to paint the roadway, try to make the words or symbols as thin as possible. It can rain a lot in February and paint keeps the water from soaking into the pavement. If you cover the whole roadway with some artistic design it can cause a very slick surface for the riders. Just ask Freddie Rodriguez who crashed out of the 2002 Tour on a huge replica of the Luxembourg flag.
Cycling to watch the race is a great way to see the riders. If you are using a route that is not part of the race course then there should be few problems. However, if you are going to be on the race course, which is definitely the case on climbs, you may be asked to either dismount and walk your bike or you may be kept from moving at all. I wish I could give you an exact time to be at your desired viewing spot before the race passes by, but because of a number of factors it is virtually impossible to say when the road will be closed to cyclists.
The race passes through numerous police jurisdictions and they all seem to deal with the event in their own way. The best advice I can give is to leave early, bring lots of warm clothes and food and convince a few friends to come along so you can have a party of sorts while you wait. Be sure to check the weather conditions. Waiting in the rain is really a drag. Oh yeah, look both ways before you venture out onto the roadway. Even on supposedly closed road, race vehicles seem to magically appear out of nowhere.
Best of luck!