You will need heirloom varieties — hybrid seeds do not reproduce true to type. Many seeds sold today are hybrids. Check to see that yours are heirloom or F2; F1 hybrids will not produce the same plant you gather them from.
Some seeds are more complicated to save than others. These need to be pollinated carefully (like corn) or are difficult to gather (like tomato). If you have never saved seeds from your garden’s plants, start with the easy ones like lettuces, orach, spinach, peas, beans, parsley and squash (that were planted far away from any other squash variety).
To learn more about seed saving, check out Seed to Seed by Suzanne Ashworth and my own Sacred Land.