It is a lot of fun to grow vegetables from seeds, and reasonably cost effective. A packet of tomato seeds may cost $2.49 (plus shipping) and you will have about 15-20 seeds in a packet. If they all germinate, you will have more tomato seedlings than you will need for a typical backyard garden. This would be very cost effective because tomato transplants cost about $4.00 each. You have just grown about $80.00 worth of transplants from a $2.49 packet of seeds. For best seed selection see pages 8-10: http://edis.ifas.ufl.edu/pdffiles/vh/vh02100.pdf.
There are many ways to grow the seedlings. You should get a quality media to start your seedlings. A quality potting mix or a germinating mix will work well. You will need containers for you seeds. I prefer to use the plastic cell packs that I buy small plants in at the garden center. They can be reused over many seasons. Peat pellets, egg cartons, small milk cartons, and even soil plugs have been used with varying degrees of success. Whatever container you use, make sure it is clean or sanitized (1 part bleach, 9 parts water) and has drainage holes. See: https://edis.ifas.ufl.edu/pdffiles/VH/VH02700.pdf.
Fill the container with a good quality potting mix, and water it well. Plant the seed no deeper that three time the diameter of the seed in the center of the container. If the seeds are from a previous season, maybe several seeds in a container will ensure germination. Water the container lightly and place a label in the container identifying the vegetable variety and date the seed was placed in it. Most seeds will sprout in about a week. Keep the soil moist until the seeds sprout.
Once the seeds sprout, move your seedlings into a bright lighted area, preferably a full sun location where they are protected from squirrels and drying winds. If you choose to grow the seedlings under light, a 40 watt shop light will do the job. Keep the light about 2” above the growing seedling and “on” for about 16 hours a day. Keep the soil moderately moist and fertilize with a half strength liquid fertilizer every other week. It will take 4-6 weeks for the seedlings to reach transplant size. January and July are the best months to grow seedlings of “warm season” vegetables. September and October are best months to grow seedlings of “cool season” vegetables.
Once the seedlings are ready to transplant to the garden (or your purchased transplants to put in the garden), water the transplants to help with their removal from the container. Water the garden soil well to minimize transplant shock. Dig a hole as deep as the transplant was growing in the container. Remove the transplant from the container and place it into the hole in the garden. Do not add any fertilizer or compost into the hole. Firm the soil around the root and stem of the transplant and lightly water to remove air from around the roots. If necessary, add a transplant fertilizer (half strength liquid fertilizer) at this time. Put a label into the ground next to the transplant so you will know what you are growing several weeks into the future.
There are some vegetables that do not grow well in containers and should be planted directly in the garden bed. Beans, corn, squash, cucumbers, watermelon, carrots, beets, onions, peas, potatoes and turnips are excellent candidates for direct planting in the soil. Wet the soil and plant the seeds no deeper than three times the diameter of the seed. Cover the seeds, wet the soil again and place a plant marker in the soil near these seeds for future identification. If you over planted the seeds, thin them early to reduce the stress of the seedlings. See: https://edis.ifas.ufl.edu/pdffiles/VH/VH02600.pdf.
Now you know how to grow your own transplants from seeds and which seeds will not do well as transplants and should be planted directly in the garden.