Farming - Artisan
Artisan - Artisan goods (wine, cheese, oil, etc.) worth 40% more
Artisan is widely considered the best profession in the game but it's probably also the most overrated one in the game. There are a few problems with Artisan: (1) generating more money doesn't give the Farmer much quality of life improvements, (2) it comes on too late to be useful unless you're trying to reach Perfection as soon as possible, and (3) it has practically no synergy with Tiller and makes that choice practically useless.
Mining - Geologist
Geologist - Gems occasionally appear in pairs (50% chance per node)
Geologist is another hidden "make more money" profession that the Farmer doesn't need when they should be focusing on quality of life improvements. While this profession might help with gift giving since a lot of the villagers like Amethyst, Ruby, and Diamond... Flowers are much better gifts, as focusing on gifts that have increased quality get a bonus 10-50% increase to the friendship gains.
Foraging - Gatherer
Gatherer - 20% chance for double harvest of foraged items
Both Gatherer and Botanist are popular among Farmers but Gatherer gets a little more attention than it should. While it's useful early in the game, it encourages the Farmer to spend time doing activities they probably shouldn't focus on -- the length of the day is already relatively short and trying to go through the Forest, the Beach, or even to the other side of a floor in the Mines to pick up something isn't always worth it. This pattern of foraging everything in sight compounds the inventory space the Farmer has during their first year but at least Botanist helps fix that once they reach level 10.
Fishing - Angler
Angler - Fish sell for 25% more gold
Again, the popular opinion goes for more profit with the Fishing professions and that feels wrong. If you decide to take Angler, please don't double down and take Angler for a total 50% increase to fishing profits, go with Pirate instead.
Combat - Defender
Defender - +25 HP added to your max health
There is almost no reason to take Defender. No monster in the game can one shot the Farmer (or come remotely close to it) so the Defender profession is virtually a wasted choice.