On the whole, it's usually south to avoid the cold in winter, and then north to avoid the heat of summer. I have done a couple winters here in CO, but the Arctic blasts can be brutal when they happen.

Resources are pretty much the same in every city, meaning there are none. Every city is overwhelmed as far as resources are concerned. Anywhere you go is the same. They'll do an intake (so they can get paid), then tell you they're swamped with clients, out of funds, thank you for coming and have a nice day. For the most part, those "resources" are a waste of time.

As for other resources... Houston has public showers (75 cents) in most of their parks so that's a huge deal. Food prices play a factor, as well as transit. Cities that are popular homeless destinations (Miami, SoCal, San Fransisco and believe it or not Salt Lake City) are 100% no-go zones. The bigger the homeless population, the more a homeless person should avoid it. For me personally, it's less about the number of homeless people and more about the number of addicts. The former will leave you alone, the latter will cause problems

How long to stay depends on a few things. Weather is first, it can be life-threatening. If a city is "cracking down" on homeless (Most cities do every March, don't ask me why), it's also time to go. And if it's just not working out, time to move on.

Wherever you go, you have to keep in mind that it's temporary. Just like everything else.