Just about every vegetable can be grown by the home gardener from seeds, but if you're relatively new to gardening, or you're teaching gardening to young people, you want to choose varieties that are practically foolproof. Pick seeds that are quick to sprout and have robust growing habits for the quickest gratification. Choose easy to grow seeds, and you can leave the fussy plants for next year.
Radishes are cool season crops that mature very quickly. They can be planted next to or among other plants in your garden bed, because they will be harvested before they take up any room that the other plants need. Plant seeds in your garden about 1 inch apart, and about 1/2 inch deep. Thin them to 2 or 3 inches apart when they are about an inch tall. You can harvest your radishes between three and six weeks after planting, depending on the variety.
Pumpkins are a fall crop in most of the country, because it needs a long growing season. Most pumpkins take about three months to mature. Plant seeds directly in the garden, leaving about 3 feet between sets of seeds. Pumpkin plants are very satisfying to grow because the vines get very large, very quickly. Mulch around the stems to help keep the weeds down during the growing season. When your fingernail can't easily pierce a pumpkin's skin, it is ready to harvest.
There are many different types of beans to grow in your garden, and all of them sprout easily by planting the large seeds directly in the soil. You can grow bush beans or vining pole beans. Beans are green or wax for fresh eating, and there is a large variety of beans for harvest after they are dried. All beans should be sown in a sunny spot with good drainage. Planting beans has an advantage for gardeners; beans add nitrogen to the soil so that next year's other plants can be healthier.
Cucumbers are one of the most popular vegetables for home gardeners to grow, and they are very easy to raise from seed. Plant seeds about an inch deep, keeping plants about a foot away from each other. Cucumbers do well climbing up fences or cages, so place supports at the same time as you plant the seeds. Keep the plants hydrated by giving them 1 to 2 inches of water per week. Harvest the cukes when they are firm and about 6 inches long. Keep picking the cucumbers every couple of days so that the plant will keep producing more food to harvest.