It is not advised to start your strawberry patch with seeds because each seed on a berry is unusual and is not "true to type." That means, you may think you're planting a berry with a specific size or color--like an oversize, red strawberry--and wind up with a completely different small, light red one. A strawberry patch grown from a handful of seemingly similar seeds could easily become a hodge-podge patch of different sizes, shapes, colors and flavors.
As nurseries and professional farmers prefer more predictable plants, those juicy strawberries you ate last summer most likely didn't come from seeds planted in the ground. Instead they probably came from a third- or fourth-generation plant that's been planted, harvested, shipped, replanted and harvested again. In fact, the only way to have strawberries grow "true to type" is through vegetative propagation.
If you decide to go ahead and plant a bed of strawberry plants from seeds, it's not too difficult to harvest the seeds as long as they are from fresh, never chilled fruit. According to farminfo.com, you can easily separate strawberry seeds from ripe fruit by following these steps:
Place fruit in a household blender with a cup of water.
Blend at high speed for a few seconds.
Wait for everything to settle (the good seeds settle on the bottom, poor seeds and pulp will float).
Pour off liquid and fruit pulp.
Scrape out seeds and dry them on a paper towel.
Seeds are a bit more difficult to find at the local nursery or farm store, as compared to strawberry plants. However, several sources are available online, including at Amazon.com, and frequently seeds are offered on eBay as well.
Strawberries are different from other fruits as their seeds are on the outside of the fruit. Unfortunately, the easily-accessible seeds and their bright red color make them very attractive to birds and wildlife.