I enjoy cycling, and as I cyclist, I'll say this - most cyclists I've seen don't deserve to share the road with cars, because they don't follow the laws. If you want to claim the right to the road, you've also gotta follow the rules. I see tons of fellow cyclists who don't honor stop signs, traffic lights, yielding laws, etc, etc. Many groups travel wider then two abreast, and many won't stay to the right of the road when they can reasonably do so.
I agree, the law says that cyclists have as much right to the road as a car, but only to the extent they are willing to follow the laws and be courteous themselves. Seems many cyclists have a chip on their shoulder that "they have the RIGHT to the road", and get all bent out of shape when anyone complains, yet they are oblivious of, or just plain don't care about, following the darn rules!
Okay, I'm off my soapbox...
I love cycling. I just got back into it after a long time of not going. Just went on my longest ride of the season yesterday, only 10 miles, but I had a kid in tow, and I'm just getting back into it, so I claim a job well done!
Don't buy walmart bikes. I tried that, and it was a major fail. I just bought a Trek 8.3 DS (Hybrid Mountain/Road bike). While not the most expensive bike you can buy, it is worlds above anything Wally sells.
Take water with you, and keep a saddlebag with at least a patch kit, maybe a spare tube, CO inflation device and/or pump. I would also recommend the Leatherman of the bike world: http://www.amazon.com/Topeak-Alien-26-Function-Bicycle-Tool/dp/B000FIE4AE
It's great exercise, and can be fun for the whole family. My wife doesn't like to bike, but she runs. So she'l run, and I'll tow the kids in the trailer. I've previously lost about 60lbs by biking alone, of course I've gained it all back, which is why I'm getting back into it now.
Enjoy it, learn the laws of cycling, be courteous to drivers and they will be courteous to you too. While you may have the right to bike down a busy canyon road, it doesn't mean it is the smartest thing to do for your safety, either.