In my experience I'd say it's easier to get to a bigger house once you're in the Woodland Bungalow or generally a house that has a few more pumpkin, plant, and pet slots than the starter house. Being able to send out more pets means that you bring back a bit more money, and more pumpkin slots of course means being able to feed all those pets! And more plant slots can also mean more money, whether you sell your plant products or use them to make Wealth Potions.

It's possible to make the jump from the starter house to an endgame house, and I think someone's done it before, but I'd imagine it's very difficult, lol