As redrobin62 said, it's much easier being homeless if you have a car to sleep in.
I speak from experience. I've been between homes a few times, a few different circumstances and places. But the car is key. I'd feel so vulnerable without a car.
On the other hand, when I was younger I also hitchhiked around the American West, and had nothing but a backpack and a bedroll. It was a long time ago; I don't think I'd want to do it now.
But, I know a young person who did something similar, about a year ago: hitchhiked, and hopped trains, or however you call it, & traveled on Greyhounds when she had money from panhandling or picking up odd-jobs. She was telling me, there are a whole bunch of people out there doing that. They run a circuit, certain cities in different parts of the country - they kind of seem to clump in groups. I don't know if that's what you want.
For me, I have never hopped a train (getting on w/o paying, often while the train is moving, for those unfamiliar with the idiom - or if I got it wrong) and I most certainly wouldn't do it without someone experienced to show me what to do. Well at this point, in my 50's, I'll never do it. But hitchhiking is pretty basic. It would probably be pretty cool with a dog.
I've never been homeless without a dog, I think that helps too. A dog and a car, you're totally set. Warm places in the winter, so you don't die of being frozen. The most recent time I was between homes for about a month, I stayed in front of the Library, where there was free wifi, while my son & dog snored away next to me. It was cool. A car can be a home.
Traveling is a good way to deal with homelessness, if you have the freedom to go. There's a lot to be seen out there, and