Once you have some cash in the bank, you can earn tons of money by selling turnips, but this relies on getting a good selling price, which may not happen offline or straightaway.
While you wait, set a course for Tortimer Island, ideally from the evening and onwards (when the giant stags appear). On the tropical island just south of the reception, you can catch rare bugs and fish, which you can store in the basket and sell at Re-Tail back at home.
To begin with, speak to Lloid the gyroid, who’s just left of the entrance and rent the net and rod. Then slowly circle the island and catch everything in sight, be it giant stags on the palm trees or fish in the sea.
You will likely need to use the advanced capture techniques to catch your prey (for the net, hold A and nudge the circle pad slightly to sloooowly creep; for the rod, close your eyes and listen to the sound of fish biting).
When it gets quiet, go in and outside the building to reload the creatures on the island and then repeat. In two island trips, you can earn enough to pay off even the most expensive house upgrade. You can also invest the funds earned here in the turnip market.