By cutting your flowers regularly and using them in gorgeous flower arrangements you are halting the natural process of seed formation. It is why the plant keeps on making more flowers - to try and get that precious seed to form. At some point in the late Summer or early Autumn you may like to stop the earnest cutting of some live blooms, and allow some flowers to develop into seeds, so that you can harvest them to store for planting next year. When you really get into seed saving, then you might like to identify flowers that you particularly like earlier on in the season, and mark them with a piece of string or a label. You can then identify the particular plant you want to collect the seed from once the seeds have formed.
To save seed choose dry, brown, seed heads on a dry, warmish day and cut them into a labelled paper bag. Label the bag with the date and the name of the type of seed. Do this before you collect the seed as it is easier to write on an empty bag. Hang the bag somewhere dry and dark for a few days to allow the seeds to dry out properly, I use my garage. Gently separate the seeds from the seedhead. Some will need less "processing' than others. Nigella for example just needs a shake of the seedpod to release the small black seeds. Calendula might need a little bit more work to separate the seeds from the bits and pieces of seed pod. It's best to remove these bits of "rubbishy" plant material as they are not needed. One way to do this is to tip the seeds fromt he paper bag onto a plain white plate, then you can see what is rubbish and discard it. If it is fine bits of seed casing then you may be able to remove them with a gentle blow over the plate. This can be a bit tricky to master, but works very effectively with the heavier seeds. When you are trying to do it with teeny tiny grass seeds it can be a bit messy! (As I know from experience!) Cerinthe are nice big easy seeds to see and they tend to drop out of the seed heads when they are ready to store. A great one to practice on.
It's best to store seeds in either glassine envelopes (easily purchased online) or small paper envelopes. Wage slip, or dinner money envelopes work very well. These can be decorated by your group to make seed packets to sell at Christmas fairs etc if you collect large numbers of seeds. It all helps to raise funds for your school garden group to keep it self financing. Your young entrepreneurs can decide on the price and how to market them, teaching them valuable business skills in a real life context.
Dont store your seed in plastic bags as they run the risk of going mouldy, if there is any moisture left in them, as they can't "breathe" in the plastic bags.
On the subject of drying seed, the latest issue of 'The Organic Way' from Garden Organic suggests using a dessicant. Trials show that the lifespan of seed is quadrupled if there is 5% less moisture in them. "Using correctly dried seed can improve its germination rate, slow seed aging and reduce the chance of mould or insect infestation during storage", producing more vigorous plants in the long run.Garden Organic suggest using rice, which has itself been dried in the oven (10 hours spread over a pottery dish at 100 degrees) and then stored in a sealed preserving jar. Use at a ratio of 6 parts rice to 1 part seed. Seed can be placed loosely in a muslin bag. Put the rice in a preserving jar and pop the seed bag in too. Seal tightly and leave for a week. Larger seeds like beans need two weeks.The seed should then be kept in packets in an airtight jar (such as another preserving jar) to prevent moisture being reabsorbed. The best place to store the jar is in the fridge.
One thing to note is that you may get a surprise next year from your "self saved seed" They may display slight variations from the original plants that you collected them from, especially if you have more than one type or colour of each plant. For example if you collect cornflower seed, and you have black and blue cornflowers growing close together in your plot you may end up with some mottled or mutlicolored cornflowers due to the cross pollination that has taken place by the insects. We think it adds to the fun on Our Flower Patch! I've had some marvelous crosses of nicotiana; lime mixed with purple, and cosmos; various shades of pink and picotee, on my flower patch this year. It all adds to the variety on your school flower patch. Sweet peas are a notable exception to this as they are self fertile, and so they will come back next year with the same flower as the flower you collected them from.
So why not give it a go!