I'll just add my two bits worth to this thread as well
Yes, suckering is pinching off the little 'suckers' that grow between the apex of a major branch and the stem. Not only will you have tastier, healtheir fruit but, the plant itself will be healthier because it will expend less energy on growing new branches. It will also be easier to stake your plant because it won't be such a tangle of branches.
Oh, and saving seeds. Just cut the tomatoe in half and as Grey said, squeeze the seeds and juice into a container or onto a plate. Then, let the mixture sit out for a few days until it becomes a little moldy. Don't worry, the fungi secrete chemcials onto the seeds that will protect them from bacteria and other fungi.
Anyway, then rinse the seeds off, pat dry with some paper towel and store in an airtight container until next year.
In about February, start the seeds indoors in flats and transplant the juvenile plants into small pots. Continue to put them up until the spring when you plant them outside.