Take the bus? Really?
Really?
I think those whom would laugh off an adventure like this with such a "practical" suggestion may be missing the point, profoundly.
I would think it obvious that anyone that would consider a cross country trip on a bicycle, motorized or otherwise isn't interested in spending a couple few hundred bucks to be trapped in a tin box, constantly assaulted by the piercing wails of squalling children, trying desperately to cling to sanity as you sit squashed between packed rows of bovine drones all sucking the same stale perfume of unwashed carcasses and diesel fumes, unable to do anything but watch another gas station roll by, bleating helplessly in frustrated anguish for four days straight with the most excitement you can possibly look forward to being struggling in vain to open your complementarity bag of peanuts...
If you even get a bag of peanuts.
A journey into the unknown as the OP suggested isn't about the destination, isn't about budget or any "practicalities" - it's about the adventure of getting there, as the title suggests. To wander the countryside, to partake of what this land has to offer, to be one with the wind, the road, the weather. To learn of yourself and the others you may meet, to not just see new places but to relish, to be part of them for no other reason than to to challenge yourself, to learn to see, to know of more than the same old stretch of endless superhighway, with it's identical rest stops dotting the countryside, with yet another McDonalds the only thing to break the monotony.
Sure it's risky, you'd better not even attempt such a trip unless you accept the fact that you may not make it, that you will spend time cold, wet and miserable, that there will be times where you feel lost and alone, that at any point you may be forced to abandon everything you've with you, that you may have no other choice than to hang your hat where you are, or to suck it up and buy that bus ticket home...
Yet what adventure would there be if there was no risk, no challenges to face?
Do it, wander this beautiful land by whatever means you can. Build yourself a motorized bicycle and if the motor quits, pedal it. If the bike quits, walk. If your feet can take it no longer you can always bail, the learning experience, the sights, sounds and smells worth every penny invested, the wisdom earned, the insight gleaned worth so much more than the price, what meager possessions you have with you, what you would stand to lose so much less than what you will gain. I've nothing but the deepest respect for anyone who would even contemplate getting off their sofa to go live for a moment, to break away from the discovery channel to go discover for themselves, to dare to dream of something different, to stop whiling away the hours, days, the years of your life playing video games to just go play.
To stop watching to go do.
If you're tethered by responsibility, obligations, time and debt - sure, getting there may be the only priority, the only option. If you're scared of change, inured to the tedious monotony, complacent comfort your only interest - then no, such a risky proposition isn't for you and that's well and good, for you... but please don't nay say for it's sake alone, don't hamper another with your limitations, ones you've chosen for yourself. Dare to dream if just for a moment, if only vicariously through others, imagining for even just a second you were not so hampered, that you too could savor the rare, exquisite taste of freedom - if you could but try, to think of what may lay in store for such an adventurer, to warn of real dangers, to appreciate the challenge, to actually appreciate the depth of experience.... who knows, you just may find you're not as trapped as you thought, you too may find a moment to break away, to go do something for a change - or at least help a guy along his way.
Two grand, a small price to pay for a life worth living, even if for just a couple of months. What there is to see, to do, to learn - these things will last a lifetime, well worth the investment even if the bike blows up, if all you have to show for it when you get there is some pictures, a handful of stories and memories rich with experience. To regret not even trying a far worse consequence than the loss of a few dollars and a bike.
Don't listen to those that would scoff, they've no idea what there is to be gained.
Any half decent bicycle & trailer will do, I would recommend a rear rack mount, chain drive four stroke for it's simplicity, quiet operation and longevity. Belts and tires wear out too fast for friction or belt drive to be the best option & much as I do appreciate the HTs, they are too loud and unreliable for the longest of trips, a rack mount easy enough to rid yourself of should it fail completely... but even that doesn't really matter, it's the doing, the trying that counts with whatever you've got, whenever you can. I wish you only the best of luck man, may the wind be always at your back, that friendly smiles & dry campsites greet you wherever you go.