I actually met up with some folks who were doing a trip across Arizona and into southern California this past January. When we came up on them on the road, one of them was biking with a bike trailer and the other was running, pushing a jogging stroller with their old dog in it. Both the stroller and the trailer had gear and some supplies. They seemed to be having a great time, with their gear just a little worse for wear, but this was in the winter, too. They had some good sleeping gear for the colder nights that time of year, but aside from that, just jugs and jugs of water, a little food, a small amount of clothes.
I guess I couldn't say exactly how it was, as I was not doing it, but they were having fun, their dogs were having fun, and they were taking backroads through some incredible areas. For me, I'd be sure to take more water than I thought I'd need, some sunblock (unless you have a dark tan/complexion), and maybe a pay-as-you-go cellphone in case something comes up in the middle of the desert and you can't ride on.