It is very easy to propagate strawberries from existing plants.
There are 3 ways in which new strawberry plants are cultivated,
1. From Runners
2. From Seed
3. By Plant Division
All three have their good points and their bad points, however by far the easiest for the home grower is to propagate from runners.
Propagating From Runners
As strawberries do not give a good crop after four or five years it is normal practice to replace the plants every three to four years. There is no need to go out and buy new plants (unless all of your current plants are either; more than four years old, or are badly diseased) as this is easily done by propagating new strawberry plants from the runners that the plant naturally sends out. However it is essential to ensure that the parent plant (the one sending out the runner) is free of all pests and diseases, especially viruses, this is because an infected parent plant will not produce a strong child plant which in turn will not give a good crop.
It is best to select healthy one year old parent plants, once you have made your selection do not let these plants bear fruit. By removing all the flowers the plant's energy goes into producing runners. A good healthy parent plant will produce numerous runners, but for best results only allow four or five to develop and then only one plant on each runner.
The runners will naturally produce a plantlet, however you can encourage this to happen by "pegging down" the runner into the soil, this is normally done in mid summer to mid autumn. If the soil has been lightly cultivated you can peg directly into the soil or you can peg them into pots (which maybe easier if you are moving the whole bed). Fill the pots with potting compost and either sink them into the ground such that their rims are just level with the soil surface, or stand them on the surface in such a way that they will not fall over.
Peg the runner down into the soil with either wire hooks or hairpins, then if you are using pots do not let the compost dry out but do not over water, once the plantlet has rooted it may start to send out another runner from the point of rooting, you need to carefully remove this so that all the energy from the prent plant goes into producing a strong healthy plant.
Four to five weeks after you have pegged down, the new plants should be ready for moving, to check that they are there should be good strong new leaf growth, which is a good sign of successful rooting. At this point cut the runners from the parent plant. If you pegged down directly into the soil use a trowel to lift your new plants ensuring that there is a good ball of soil around the roots. No matter which methods of propagation you used the new plants should be planted into their final location immediately (see growing strawberries for details on how to do this). To get a better crop in the second and subsequent years do not allow these new plants to flower during their first season.
If you follow these guidelines for propagating your own strawberries then you should have a good healthy supply of strawberries for many years.
Growing From Seed
Starting strawberries from seed is not hard to do, however getting a "true" plant that will bear fruit is not always guaranteed due to the way that today's varieties have been cross bred. This means that attempting to grow strawberry plants from seeds that were collected from a purchased strawberry will likely not produce true plants. In other words, the plants grown from seed will be different from the plant that produced the seeds (sometimes substantially so). The seed-propagated strawberry plants will exhibit genetic traits of the grandparent plants instead of the desired parent plant.
The wild species of strawberry to not have this problem and will produce a "true" plant from seed, however their fruits are much smaller that the cultivated varieties and as such most people would not recognized them as strawberries.
Many varieites of strawberry will under the right conditions produce lateral crowns at the base of the strawberry plant, it is these lateral crowns can be divided and replanted to propagate strawberry plants. The main draw back to this method is that it requires a lot of skill to avoid killing off the parent plant and for teh home gardener it is more trouble than it is worth.