Pokemon Trainers, Coordinators, etc. all have free accommodation and food when in use of a Pokemon Center.
Ash's mother, Delia, owns a Pallet Town restaurant. She may be sending him money, or paying for everything that's been sent home by Ash. And of course, there's the mystery of his father and what he does.
Misty owns a Gym, and has relatives working there when she isn't present. She is likely well off. Not to mention the possibility of her parents being financially achieved.
Brock is similar to Misty, in that he has inheritance to a Gym as well. Being the one with the most items on his person, he may have the most money on hand out of all of the travelling group.
Tracey had a short-lived time travelling, but his parents likely have him checked off as paid enough to travel alone. And then there's the chance that he works as a pokemon sketcher for a profit.
May and Max fall under the same category as Misty and Brock, since their father owns a Gym. Electronically sent money to his kids, or electronically sent bills to their parents, can set them pretty well for their time on their journey.
Dawn's mother, Johanna, was a successful coordinator and has been shown to be quite wealthy. It's likely she takes care of Dawn's expenses, and, just like Ash, Dawn's father may play a role in money troubles.
Basically, I think they have an amount of money on their person, but not enough to get by for their journey. They must rely on relatives, is what I'm saying.
Either that, or the writers don't think they need to care.