The single most important natural product for lawn care is corn gluten. It is sold under a variety of brands, and most garden centers carry it. It is a weed preventer, though you might be a little late now to have it down in time for this year. But it's also a fertilizer. There isn't a lot you can do about broadleaf weeds in your lawn with natural products. Get a good weed tool and remove dandelions by hand. Mow your lawn tall, which will help it outcompete weeds.
There are many natural fertilizers that you can find in garden centers, and compost spread out over your lawn also is a great fertilizer and general strengthener.
You probably won't have too many insect pests, but grubs are a common problem. You can use nematodes to control them. Expensive, but can work pretty well if used exactly as directed. If a local garden center doesn't carry them, you can mail order from gardensalive.com.
There are more products out there, but the main strategy in natural lawn care is to maintain a healthy, strong lawn. You will have more weeds than in a conventional lawn, but in a healthy turf, they wont' be too obvious. So keep it well fertilized as i described above, mow it high, and it should do well.