Of course you can just buy potted plants at your local gardening center or hardware store, but if you want to grow a large number or want to try different varieties that might otherwise be unavailable, starting from seeds is the way to go.
In this video from This Old House, Roger Cook explains his technique for planting seeds. The most important part is the soil; the dirt from your backyard is far too heavy for seedlings. Cook recommends a mix of sphagnum moss and vermiculite to make a light and spongy potting soil, and he premixes it with water so that it’s already saturated when it goes into your container of choice. As the plants grow, you want the soil to remain moist but don’t drench it with a watering can; just use something like a bottle sprayer.
The seeds should be placed just barely below the surface—Cook suggests that the hole you dig is only two to three times the size of the seed. They don’t even need sunlight at first, and you can use an electric heating pad if they need some extra encouragement. Once they spout, Cooks uses a grow light to give them 14-16 hours of daylight, and before transplanting them to their permanent spots in the garden, he builds their resilience by setting the new plants outside for two or so hours at a time over a few days. Watch the video for a full explanation on how to care for your seeds and start your garden.