This is a very simple guide for finding clean water.
Clean Water Near Homeport Town
This is the first method for obtaining clean water
Near the town there is a mountain with a Greek Temple, right next to it there is a pond with clean water, press E (or your "use" key) on the water (or in the water) and your character will drink the water.
You can also fill your waterskin with this water but beware your waterskin will slowly drop water over time and if left unattended before you know it there will be no water left!
How to Find Clean Water Anywhere on Any Island
This is the 2nd more practical way to obtain water early on
To begin you must press the "X" key (or your keybind to go prone). This will make your character go prone on the ground. Click your left mouse button and a minigame will appear.
This minigame uses timing, so you must time it in the right zone three times.
After you have done this water will begin to squirt out of the soil. Press your "E" key to drink it (or your keybind to "use").
To do this again you must move into a different zone with water in the soil! There are plenty of areas on the Homeport islands (or any island) to do this!