The issue is are they hybrids or heirlooms/open pollinated varieties. If they are hybrids then the seeds you save won't breed true. Hybrid seeds in the packet are the result of manual cross-pollination of at least 2 parent varieties. The seeds saved from those hybrid plants once grown, will revert to the parent stock or some strange combination of them.

If your seeds planted and grown are heirlooms or open-pollinated plants, not hybrids, then yes you can save the seeds and re-grow them. Depending on what the plant is - self-fertile like tomatoes or insect pollinated like most flowers, they may or may not grow true. In some cases, like hollyhocks you will still get lots cross-pollination by insects so your saved seeds may change color etc.

It all depends on how much you care whether or not the seeds you saved breed true or not. Many don't care, many do. Some research on cross-pollination will explain this in more detail. ;)

Dave

when I buy seeds from the store I try to get heirlooms if I can. I also look at the label and see if it says hybrid. If it's a variety I really want, I buy the hybrid anyway. If the pack doesn't say hybrid on it, it may not be. I get out seed catalogs or use Google to search for information on any specific variety to find out what I can about it.
You'd be surprised how many are not hybrid and seeds can be saved. Herbs are some that often can have the seeds saved from. Many flowers also can be harvested for the seeds, zinnias, merigolds, four o'clocks are just a few. Veggies, such as certain squahes, okra are some I have saved seeds from and grown.
Anytime you have a question, look it up, or just try it anyway!