Easily gather all the Stardrops! These locations/Tasks are in general order of occurrance so you can follow along easily as you play the game and go through the events. There are currently 7 known ways to recieve Stardrops in-game without the use of codes.
Stardew Valley Fair
The Stardew Valley Fair occurs on the 16th of Fall every year.
A Stardrop will be sold for 2,000 tokens at the Stardew Valley Fair. This is the easiest one to find. The Fishing Minigame is a handy way to gather these tokens. So is placing first place in the showcase for two years (Rewards 1000 tokens).
The Secret Woods
A Stardrop can be obtained by entering the secret woods and giving Old Master Cannoli a Sweet Gem Berry. Do this by chopping down the large log on the far left side of the forest map below your farm after upgrading your Axe. Inserting a Sweet Gem Berry into the mouth of the statue on the left will cause the statues' eyes to light up and you will recieve a Stardrop.
Into the Mines
Reach level 100 in the mine. There will be a chest containing one Stardrop.
A Stardrop can be bought for 20,000 G from Krobus, who resides in the sewers.
The sewers can take a while to get to as you need to first donate a total of 60 items to the Museum in order to get the Rusty Key, which you use to gain access to the sewers.
Increase your spouse's heart level until it is 12-13 by giving them gifts. Talk to them, and the he/she will give you a Stardrop as a gift.
If you catch every type of fish in the valley, you will recieve a letter from Willy congratulating you and gifting you with a Stardrop.
The Museum Collection
Collect and donate all 95 items to the Museum.