Here's how to grow your own tomatoes from tomatoes you eat:
1. use heirloom seeds: if you can, make sure you have an heirloom tomato. That means that the seeds are fertile and tends to be related to its parent plant (which you are eating), plus you'll be preserving the natural tomato genetic diversity... and growing super tasty tomatoes. Hybrid tomatoes like sun golds, sweet hundreds/millions, and early girls do not have seeds that will produce a tomato exactly like the parent tomato, though they are mighty fine tomatoes anyway, and worthy of buying seeds for every few years.
2. pull out seeds and put them in a water bath: before you chew and swallow the tomato piece, use your fingers or a spoon to pull out the little seeds (avoid pulp) and put them in a little cup to allow them to ferment in a water bath for 2-3 days.
3. strain and dry the seeds: pour the water with seeds in it into a fine mesh strainer (or cheese cloth), and then allow them to dry as is or wipe them onto a paper towel to dry
4. save for next season: once dry, store the seeds in a clean, dry and dark location. I usually pull them off the paper towel and put them into little plastic baggies with labels (when you have 30+ tomato seed varieties it can be easy to mix them up if you're not careful!)... and then store them in a closed glass jar in a closet.
5. get ready for magic next season! I usually start my tomato seedlings indoors in February, transplant them to larger pots in late March, and then plant them outside in mid to late May. Last year my tomato plants were small and purple and spindly in March and April... and finally after fussing with their light, water, etc., I realized that they didn't have enough nutrients in the soil I used to sprout them in. They promptly grew into a tomato jungle after I added organic tomato/vegetable food. And voila! After that initial scare, we now have a wonderful bounty of delicious heirloom tomatoes every day! It's amazing how responsive plants can be.