Luisa, to get more of the same daylily, just wait. In a few years, your small planting will become a clump that you can dig up, wiggle/cut/pull apart. You can replant the fans, sell or give away extras...
To make a brand new daylily, take the pollen from one flower and put it on the pistil (the girly part) of a different flower of the same chromosome count (same ploidy). I think there are some exceptions to the ploidy rule- more experienced folks than I can give more info.
After you pollinate a flower, just wait. A pod should form (the base of the flower will swell) and grow. Once the pod matures (6-8 weeks tops), it'll maybe change colors. It will start to split. Collect the seeds. Dry them for a couple of days. I leave mine on the countertop in the kitchen.
Put your dry seeds in baggie and stick them in the fridge so they will think it's winter (30 days?). Then they will need moist refrigeration. We all have different methods for this. I usually just put one drop of water in the baggie, some folks dampen a small piece of paper towel, even a damp cotton ball might work. Leave them for another 30 days. The cold treatment is called "stratification" and fools the seeds into thinking they've gone through winter/early spring so they will break dormancy and grow. Many seeds will sprout in the fridge.
Plant the babies in some kind of growing medium and make sure they have good lighting. They can stay in little pots or go in the ground. I wouldn't plant seedlings in the ground here after late summer. They need time to grow to make it through the winter. As you're in zone 8, your timing would be different.
Okay gang, what did I forget?
