Many years ago, when playing FoMT, I discovered that I could take my Basket to the Races and winning Medals, purchase a total of 39 brooches there, placing 30 in the Basket, 8 in my large rucksack and 1 in my hands, leaving my Basket at Rose Square until I could throw the brooch in my hands into the shipping container at home.
This provided sufficient wealth to allow me to purchase every possible upgrade and as many seeds as any farmer could desire before the end of Spring in the first year.
When More Friends of Mineral Town was released, I discovered I could use the same exploit in that game. Furthermore, racing was fairly predictable up to a point and it helped immensely that one could save the game directly before one entered Rose Square, making it easy to reload if the wrong horse won.
One can use this trick up to a point in Harvest Moon Boy & Girl as well. I have not tested it with the Basket yet because the races are wildly unpredictable and I grew tired of reloading the entire day. As you can save your game only at the point of going to bed at night, you have to replay the entire day if you wish to replay any part of the day, including the Races on 18 Spring.
If you can bear it though, prior to the point where you can win enough Medals to be able to purchase the Power Berry, you can trade your Medals for Earrings at the cost of 33 Medals each. Earrings ship for 2000G. As the price of any ticket is 50G, even if you bet on a horse with odds of 1 to 1 in the First Spring to win only 33 medals on a bet of 33 Medals, you can purchase Earrings and ship them for 2000G, giving you a profit of 350G. If you win on a horse at better odds, obviously you can buy more earrings and make a greater profit.
The problem here is that the results really are completely random and not even related to ordinary probabilities. A horse going off at even odds can lose to a horse that goes off at 13 to 1 odds. It happens frequently! As previously stated, if one could reload at the point of entering Rose Square, it wouldn't be so bad, but when one must replay the entire day, it becomes an exercise in frustration.
Your best chance to win enough Medals for the Power Berry is when you first race your own horse. Your first opportunity is the Spring of the second year. As it will be the horse's first race, the odds will be very high. If you can win at those odds, you should be able to purchase the Power Berry.
All other prizes, apart from the Earrings, are not worth the investment. A ticket costs 50G. A Truffle costs 12 Medals, making its price higher than its shipping value of 500G. The other Jewelry items, although they COST more, do not ship for more than the Earrings. All jewelry ships for 2000G.
When the Lumber is offered AFTER your purchase of the Power Berry, you should be at a point in the game where you never will need that quantity of Lumber again. By the end of the first year in fact, you probably will have all building upgrades including the Hothouse. It therefore always is better to exchange medals for Earrings and then ship them. Of course, once you have lived through a year or so, you probably won't need the profit from ANY shipments and therefore may wish to give away the jewelry instead of shipping it. That is your choice, of course.
You always could opt to trade your Medals for Bodigizer XL or Turbojolt XL but by then, you should have a full kitchen and be able to make food items that work almost as well and cost far less...