Watermelon doesn’t ripen any further once it has been picked, so it's worth your while leave it on the vine until it is ripe. After harvesting, a melon will get softer but not sweeter.
When harvesting the melon, make sure that it is cut from the vine instead of being pulled.
Since a ripe watermelon consists of more than 90% water, the heavier it is, the better. Turn the watermelon over and look at the underside. It should have a creamy yellow spot from where it sat on the ground and ripened in the sun. If the spot is greenish-white instead of yellow, leave it on the vine a little longer.