Watermelons, Cantaloupe, Honeydew, even most exotic melons such as Tiger Melons and Canary melon are extremely easy to save seeds from, they're a great place for novices to get started.
The second method is to place the seeds with pulp still cling to them, or them still clinging to the pulp in a jar of room temperature water. The jar should be full, not half full or half empty. Let them sit in the water for 3 to 4 days during which time viable seeds will sink to the bottom. Those that float should be discarded.
Once you have removed viable seeds from the water discard the pulp and bad seeds and place the good seeds on absorbent paper towels to dry off a tad.
Removing Excess Sugars
Regardless of whether you use Method 1 or 2 you still should still get rid of excess natural sugars. Melons have a very high sugar content which is good when you're gonna eat them, but not so good when you want to store the seeds.
Natural sugars facilitate microbial activity - bacteria, mold, fungus which can kill off your seeds while in storage. You want to remove as much of the sugar as possible. The most common and easiest way to do this is with dish detergent.
Gently rub a modest amount of dish detergent on the seeds and then rinse a second time. Try to avoid detergents that are high in chemicals, an eco friendly detergent is best. Dawn works for me but there are others that some feel are better See: Environmental Working Group Dishwasher Detergent Ratings
Storing the Melon Seeds
Once you've rinsed off the detergent you now want to dry the seeds prior to storage. Too much Moisture in the seed can cause them to rot during their storage. Place them on absorbent paper towels and allow them to dry out for a day. I like to blow dry them a tad with a hair dryer on a low cooler setting, although this is not absolutely necessary it does help.
Put the seeds in a airtight sealed jar or zip lock plastic bag for winter storage. The reason for the jar / plastic is so you can monitor the humidity. If you notice any condensation on the inside of the jar - then there is unwanted moisture within the seeds. They could potentially grow mold or mildew.
Silica Gel works well for helping to keep stored seed dry. Silica gel is beads of amorphous silicon dioxide. Clear silica gel is packaged with many food and non food products and labeled 'do not eat.' Placing these in the jar or baggie with your seeds is a good idea.
When properly stored Melon seeds will be viable for up to 3 years. Allthough their viability will wane exponentially.