I don't recommend this as your first multi-day tour. The trip you've proposed has hazards which will test even the most experienced riders, and you'll need to plan very carefully if you want to make it to Tucson alive.
There are some things I'm quite surprised no one mentioned about this particular route:
First, there's basically one route for cycling from LA to Tuscon, and it's almost exactly the same route for driving from LA to Tuscon. I-10, CA-111, I-8, I-10. In all you'll be spending about 300 miles, the majority of the trip, cycling on freeway. (To get an idea of what it will be like, see this video.)
While cycling on the freeway isn't necessarily any more dangerous than your average city street, it can be if shoulders are narrow in places (most often old bridges), or if there is construction, or if you have to swerve to avoid debris.
And you will be avoiding a lot of debris. Freeway shoulders collect lots of stuff that will be complete hell on your tires. Beyond the goatheads that are everywhere, you will run into the steel wires that come out of disintegrated truck tires. For this ride you will need the most puncture-resistant tires you can get. Continental Gatorskin, Schwalbe Marathon Plus, etc. Anything else, and you will be changing flats more often than riding. And even with puncture resistant tires, bring plenty of extra tubes.
I don't recommend cycling on the freeway at night. Even with lights, you won't be visible for long enough for drivers to figure out what you might be before they pass you or accidentally drive right into you. Plus, from experience I can say that your headlight probably isn't strong enough to light your way. Several of the scariest situations I had while touring were night riding and not knowing if I was about to run into road debris because I was going too fast for the distance of my headlight beam to cover.
Plan to stop early each day, well before sunset. Fortunately there is a whole lot of nothing out there, so finding a campsite won't be difficult. Typically you want to start riding around first light and stop to avoid the worst of the heat of the day in midday, perhaps riding some more in the late afternoon.
Another issue with freeway cycling is construction. You need to check with Caltrans and the Arizona DOT for any construction along your route, as a construction area may not be passable for cyclists depending on the work being done. For instance, the area where the I-10 bridge collapsed last July is not passable for cyclists at this time (though it's not on your route, and a short detour exists). Check for construction a few days before you start the tour, as Caltrans doesn't usually make this information available more than a week in advance. And note that for most of this route there are no alternate routes, so if there is construction going on it may mean a 100 mile or more detour for a cyclist, or rescheduling the trip.
Second, there are very large stretches of this route with no services and no towns to speak of. Do not let Google Maps fool you; some of the "towns" shown on the route are no better than wide spots in the road, and some of them can hardly be said to exist at all.
I would be carrying at least 8 liters of water at all times whenever possible, and plan to drink it all in a day in moderate weather. If you do this in the summer, which is ill-advised, figure on 12-16 liters of water each day. Use a hydration system (e.g. Camelbak) so that you don't have to cycle one-handed on the freeway or throw your neck out while drinking. You need to drink before you feel thirsty. If you feel thirsty, you're already well dehydrated, and if you get a headache, stop immediately because you're far gone. Plan to stop anywhere that might have water, such as the freeway rest areas, to top up your stores.
I would also carry three days worth of food at all times. While I don't expect any cities with lots of services on this route to be more than two days apart, you want to have something in reserve in case something unexpected happens.
Third, the desert weather in summer will be extremely hot. This is not something you want to do without prior training or experience. Until you know exactly what you're in for, I strongly urge you to reschedule your tour to a more temperate time of year.
A few other notes:
Skip the stationary bike; it will just be a waste of money. You live in LA; there's so rarely bad weather any time of year that you can just get a used road bike and go hit the cycle paths that run all over everywhere.
When you train, start small and work your way up. If you do a 10 mile ride and at the end you feel exhausted, then rest up and do another 10 mile ride in a couple of days. At the end of a few of these, you will not be exhausted, but will feel like you could have gone farther. Then bump up the mileage on your next ride. Repeat until you can do 50 miles or more without thinking too hard about it. Plan your rides as loops that you can cut short if necessary.
After you can do 50 miles a day and feel like you can do it again the next day, find a two or three day ride to go on, so you can get some experience with the outdoor and camping aspects. For instance, you could ride from LA to San Diego (and take Caltrain back); this will be a two-day ride at the pace you will need for your tour.