Putting the fruit into a blender, adding a cup of water and pureeing for no more than five seconds will remove the seeds from the fruit. Let the mixture sit for two to three minutes to allow viable seeds to sink to the bottom of the mixture. Drain the pulp and immature seeds from the viable seeds carefully and rinse thoroughly. Then dry the seeds on a plate for two weeks or until completely dry.
If you are concerned with damaging the seeds by using the blender, you can use a sieve to remove seeds from the fruit. Press fruit through the sieve, rinsing the excess pulp from the seeds while still in the sieve, to eliminate lost seeds. That method is more natural, but it is more difficult to separate the mature seeds from immature seeds. Dry separated seeds on a plate to store for next season's planting.
If your climate allows, you can air dry the fruit, with the seeds intact, by setting the fruit on a tray in the sun for several days. The heat of the sun will dry the fruit, and you can then rub the seeds onto a plate or into a marked envelope to store. You can also use a low heat food dehydrator to dry strawberries, then remove the seeds.
Another way to plant a new crop of strawberries is to plant small over-ripe strawberries, or slices of strawberries. That is the simplest way to grow strawberries from previous harvests, but there is a drawback. That would be having to save the fruit whole to plant when it is time, to prevent rotting. With any method, choosing heirloom or first generation ("F1") hybrid seeds will produce new harvests from saved seeds.