That would be a truly epic trip. I wish I could do it, and if I could, I wish I could do it twice; once with 13 of my closest friends and once by myself.
My best guess is that it will take you about 10 years of planning, gaining experience, and applying for a permit to be fully prepared for a trip like this. It will also probably end up costing around $10,000 once you cover all the standard expenses and buy all the necessary equipment.
That being said, it can be done. People have done it before and been very successful. Other people have died. Seriously.
It's a very serious undertaking, and is in no way, shape, or form similar to the kind of adventure you go on with a motorcycle or camping without a gun. The river is an uncontrollable variable that will eat you up and spit you out, if it feels like it.
My suggestion would be to do some of the other desert floats in Utah before the Grand. There are plenty runs up there that are much easier to manage. They have easier rapids, they're shorter, and the permit system is much more lax. The scenery and solitude is very similar to the Grand.
Check out this website: American Whitewater - National Whitewater Inventory
Click on any state to see all the rivers in each state and read about some of the other desert floats. Do a few of those, buy the gear (raft, cat, kayak, etc), then scare the shit out of yourself a few times on big water (Try Cataract Canyon at high flows), and then giv'r.
Another good resource is my favorite guide book, "Whitewater of the Southern Rockies". It's a huge book full of information about every single river and creek south of Montana. It has in-depth beta on all the desert floats. If you read the book, you can find several isolated 5-10 day trips over class I-III (easy) whitewater. There's even a run called "Desolation Canyon" - does that sounds like what you're looking for?
If you're serious about this, welcome to the addiction, we're glad to have you.