Melons, such as cantelope and honeydew, are actually quite different from watermelons, as watermelons don't ripen any further once picked from the vine (who knew!?).
For this reason, watermelons should be stored in the refrigerator once you get them home. Consider this when you're looking at those beautiful 10 lb. giants at the store!
Storing them in the fridge will keep the flesh from becoming dry, mealy, and fiberous. Once cut, make sure to cover the open sides with plastic or bees wax wrap to prevent the melon from absorbing flavors of other things in the fridge. Or whenever possible, cut up the entire melon and store in a reusable container to make it an easy snack to reach for.
Whole watermelons can last in the fridge for up to a few weeks, but once cut, try to consume them within 3 days.
Other melons, such as cantelope and honeydew, can be left on the counter to ripen further. A ripe melon will give a little when pressed on the stem end, and will produce a sweet smell from the blossom end. Note that melons, similar to other fruits, give off ethylene gas, so keep them away from other fruit to prevent rapid and over-ripening.
Once ripe, store the melon in the fridge, or cut it open for eating! If you're only able to cut up half, leave the seeds in the other half and cover it with plastic or bees wax wrap. Make sure to eat it up within a few days to get the freshest taste.
*If you're having trouble using up your melon, or it doesn't have a flavor or texture you love, consider making it into a refreshing drink: Agua Fresca / Melon Sa Malamig