Firstly, you should avoid saving seeds from 'F1 hybrid' varieties (this includes most fruit and veg from supermarkets), as their seeds are likely to be sterile or produce plants that are different from the parent plant. Seed packets should indicate whether seeds are hybrids. Open-pollinated varieties can be purchased from Real Seeds.
Some seeds are easier to save than others. Lettuce, tomatoes, peas and beans are a great place to start if you're a beginner. This guide by Real Seeds explains how to save seeds from a variety of plants. You can find more guidance at seedsavers.org.
Recommended documentary 'Seed: The Untold Story'
In the last century, 94% of our seed varieties have disappeared and a handful of chemical companies control the majority of the world's seeds. Seed: The Untold Story follows passionate seed keepers protecting our 12,000 year-old food legacy. We recently watched it and would highly recommend it. Watch the trailer.