The short answer: KLR650
There is no "best" bike for a cross-country trip. The details of where you plan to ride will militate in favor of one bike over another. How much off-road do you plan to do? all unpaved, as on the Trans-America trail, or a mix of paved highway and unpaved exploration? How gnarly do you care to go? How much time do you have?
By the same token, your choice of bike will dictate to some extent the kind of trip you can take. The bikes suggested by others are all worthy candidates, depending on your pocketbook and the kind of trip you envision. Other possibilities as yet unmentioned are the Suzuki DR650, the KTM 640/690 singles and 950/990 twins, the BMW F800GS, and the venerable R100GS boxer.
The more highway you want to ride, the better the twins will serve you. Conversely, the worse roads and trails will favor a single, the lighter the better. The more gear you carry, the more you will be limited to better roads and bigger bikes. OTOH, if you plan to spend a bit of time in interesting areas, you can always make camp for a few days and explore on an unladen bike.
There are a lot of other variables to consider, but on balance, unless I misread your view of the trip, your best choice will be a 650-class bike. I like the DR650 best, but since you want to carry a lot of gear, that means the F650 or the KLR. The F650 is the bike that first got me interested in dual sports, but I never really liked it well enough to want to buy one. It is pretty good on the highway, and the fuel injected engine gets great mileage. The KLR is quite a bit cheaper, not bad on the highway, and has a ton of available accessories..