You of course have your melon. So, cut it open and get the seeds.
Now you have a mass of pulp and seeds. There may be seeds still stuck to the strands than used to run down the middle of the melon. If so, just pick up the strands and let them hang. Run your hand down the length, stripping off the seeds as you go.
What you are left with now is a bit of wet seeds and a little pulp. At this point I use a strainer and put all of the seeds in it. Run water over the seeds and swish it around. The strainer will help scrub off most of the rest of the pulp leaving you wet seeds with just a little bits of stubborn pulp clinging on. As you dry them the little bits of pulp will be removed. What I like to do is use paper towels for this. The seeds will stick a little at times but the pulp really likes to hang on. As pulp dries it gets sticky. As seeds dry they slip.
Now you have dry seeds. Well, mostly dry. You don't want to stop just yet. If you run your fingers through them they will feel cold. This is due to the moisture still in them that will rot them if not removed. I've stored too many seeds when I "thought" they were dry to make that mistake again. Pick through them again looking for any hidden pulp and remove it usually just a film.
After the film is gone and you think they are dry enough to store you might let them sit out for a day on a paper plate to fully dry. They will feel a little cool still but not cold to the touch.
I use plastic bags or canisters but if yo want to use paper for storage it breathes a little better and will help pull out moisture just in case some is left. Check the bags for problems for a few days. If a problem starts you can usually save most of the seeds and remove the problem.
Next season you can plant.