I've been using this method of saving tomato seeds for awhile and I thought it was worth sharing. Unlike most seed saving methods, this is not as cut-and-dry as leaving the seeds on a paper plate to dry out. This is a process of fermentation and does a good job of eliminating the goop as well as many tomato diseases that could be present.
Step 1) Select the ripest tomato that you can find. Choose a tomato that expresses all the traits you would desire to grow again and again. Maybe it's a big, juicy heirloom or perhaps a roma variety; your call.
Step 2) Prepare your tools. All you need is a clean container, a spoon, plastic wrap, a sharp object and of course, your tomato! The idea here is to kill any diseases that may be present on the tomato seeds and remove any goop and grime; no point in introducing any more, right? Wash your hands.
Step 3) Cut or pull the tomato in half. If it is a small tomato, I find it easier to cut it from pole to pole. For a larger tomato, you may get better results by cutting it along it's equator.
Step 4) Scoop out the seeds and drop them into your container. Don't worry if there's a lot of goop, the fermentation process will dissolve most of it.
Step 5) Fill the rest of your container 3/4 full with cool to luke-warm water and cover it plastic wrap.
Step 6) Poke one or two small holes in the top. I'm using a toothpick.
Step 7) Leave it in a warm place. I'm keeping mine on top of the refrigerator. A sunny windowsill also works. For the next three days, stir the mixture daily. Pretty soon, you'll have something like this.
Step 8) Once fermentation is complete, you will likely have a layer of scum floating on the surface. Use a spoon to scrape it away and pour the rest of your seeds in a colander and rinse with water.
Lastly, spread the seeds evenly on a paper plate as usual and let them dry until you can snap one in half or until they move about freely without sticking to each other. This will probably take a few days to a week. Once they are dry, save them in a paper packet and store them in a cool, dry place.