I start by cutting open the melon and harvesting the seeds. This is easier with a melon where all the seeds sit in the middle like the White Sensation Melon or French Orange Melon I started with. This example happens to be for a Yellow Dole Watermelon. I'm collecting seeds as I go.
Then I rinse the seeds off to remove any stickiness or fruit residue.
Next I let the seeds dry for at least three days on a paper towel. I change the paper towel every day.
When they're dry enough that you can't bend the seed, I put them in a labeled bag and store them in a cool, dark place with low humidity.
I got a cute box from Papersource and mini file folders to organize seeds that I've bought and harvested on my own.