There are seven Stardrops to find.
• Reach Floor 100 in the mine, this will likely be the first one you obtain.
• Purchased for 2000 coins in the Stardew Valley Fair (16th of Fall). Note: Green has a minimum 75% win chance (up to 95% depending on your random luck for the day) so use this to gain the required tokens quickly.
• Get married. When you reach 13/12 hearts (that isn’t a typo) your spouse will gift you one. Note - you can buy the Bouquet from Pierre to start dating when on 8 hearts with a marriage candidate. This unlocks the ability to reach 10 stars with someone. Then you get the Mermaid Necklace (5000g), which is required to propose, from the vendor on the right hand side of the beach (over the bridge that costs 300 wood to repair) when it is raining. You also need the first house upgrade.
• Purchase the rare seed from the traveling merchant (NE of wizard tower). Grow it during Fall (early, as it takes 25 days) and give the Sweet Gem Berry to the statue in the secret woods north if the Wizard’s tower (accessible using a steel axe)
• Donate 60 artifacts to the museum to get the sewer key. Buy one from the vendor there for 20,000 gold.
• Catch every type of fish (required for the Master Angler achievement anyway) and Willy will mail you one.
• Complete the entire Museum collection (Books, Artifacts and Minerals).