I found the most successful and easiest way to start seeds is the paper towel in plastic ziploc bag method. The concept is simple. You take a wet paper towel and place your seeds onto it. You then cover the seeds by folding the rest of the paper towel over them. Then you place the paper towel containing the seeds inside a plastic ziploc bag. Place in a warm dark place and check seeds every 5 days.
I was able to quickly germinate tomato and pepper seeds in 5 days. The main advantage of this method is that you save yourself from re-potting twice. Normally you would start seeds in a seed starting tray. These are 1.5" wide by 3" deep cells in 50 or 72 cell counts. Once your seedlings reach a certain size you'd have to either plant them in soil outside or re-pot them into a larger container if it's still too early to plant outside. Germinating in a paper towel allows you to take newly germinated seeds and place them directly into a larger pot like a 4" container. This would remove the need to mess with the seed starting trays altogether.