There are many advantages to starting plants from seeds. It is much cheaper and you have a much larger variety of seeds available than you do plants. The downside is a fair bit of work is required and it takes quite a bit longer to get a mature plant.
Use fresh potting medium and pots. The biggest problem most of us have with starting seeds is ‘damping off’ which is caused by a fungus. When this fungus is present, seedlings wilt or do not even germinate. You can also use bleach to clean out your pots if you going to reuse some.
The less you have to move a seedling the better. If you must separate and replant seedlings loosen the soil with a pencil or skewer. Grasp the seedling by a leaf, not stem or root. Drop seedling root into an already prepared hole in its new pot. Seedlings can recover from leaf damage but not root or stem damage.
As a general rule of thumb plant seeds about twice their depth down in the soil. Some need light to germinate, some dark. Be sure you know which your seeds need, most need some light.
Soda bottles with the bottoms removed, clear plastic cups and take out containers all can be used to make green houses for your seedlings. Do not use saran wrap, it does not let air through. Be sure to only use clear containers for covers. Your seedlings need their light.
Some seeds need a period of cold before they will germinate. Use some sphagnum peat that you have soaked, then wrung out so it is damp but no longer dripping when you squeeze it. Mix your seeds into this mix and place in a clear plastic food container. Lunch containers work great. Leave outside for 2 or 3 warm, sunny days. ( Seeds will take up water and swell. ) Now place the bag of seeds and peat in your vegetable bin in your fridge for 3-8 weeks depending on how much cold time your particular seeds require.
Seeds may be marked F1 or Hybrid. This means that they are the result of hand pollination between two parents. Seeds saved from hybrids may not be true.
F2 seeds are the seeds of F1 or Hybrid plants and have been self or inter-pollinated.
If collect or are interested in heirloom seeds check out Seed Savers Exchange
Seeds should be stored in a dark, air tight container, just above freezing. Your refrigerator will work well.
If you are planting seeds from plants that prefer dry conditions, like cactus or herbs use a vermiculite/perlite mix rather than soil.