There are a few ways to get ancient seeds:

 - Donate the Ancient Seed artifact to Gunther at the museum and he will give you a plantable packet as a reward. He'll also give you a crafting recipe to turn any other Ancient Seeds into the plantable version.
 - The travelling cart occasionally sells them for 400-700g
 - Putting Ancient Fruits into the seed maker will give you seeds (though you need an ancient fruit seed to start your ancient fruit plantation)
 - Other plants have a very low chance to give an ancient seed when put in the seed maker

The Ancient Fruit plant is unique in that in grows in all three growing seasons- spring, summer, and fall. It takes a whole month to grow and produces fruit every 7 days, so it is best to plant it at the start of spring or in a greenhouse.