I decided to start seeds inside. A plant costs a few dollars; a packet of seeds costs about a dollar for a few hundred seeds. So I bought packets of all of the vegetables I want to try to grow this year: tomatoes, carrots, radishes, basil, oregano, thyme, cilantro, lettuce, beans. I'd also like to try cucumbers.
How do you start a seed growing? Germination. From what I can tell, if you put a seed in a moist, dark location at a reasonable temperature, it should germinate. The picture at the left is a Jiffy germinating kit. There are 72 individual cells, and it comes with 72 little compressed pellets of peat. You add some water to each cell and the peat expands. Stick a seed or a few seeds into each cell at the proper depth, add some water to the bottom, cover, and wait.
Rule of thumb seems to be -- bury the seed 3-4 times the size of the seed.
The problem with this kit is that the seeds all germinate at different times, and once the seed has germinated you are supposed to remove the cover. When a few different types of plants had germinated I removed the cover and the rest seemed to do OK.
Here's a bit about germination:
Washington State University extension