Summary: Baking with a sourdough starter instead of yeast, and making your own sourdough starter from scratch, is often thought of as a lengthy and intimidating process, but creating your sourdough starter from scratch is really straightforward and all you need is flour, water, a few short minutes everyday, and within about a week, you will have a nice active sourdough starter to bake with.
Many traditional recipes for making a sourdough starter call for a cup of flour to start, and another cup for the next seven days, discarding half of the starter everyday. This makes for a lot of wasted flour, so in order to keep our precious flour supply for actual baking, we are making our sourdough starter using a much more modest amount of flour, just one or two tablespoons of flour at every stage. This is why we refer to our sourdough starter as a small sourdough starter. You still need to discard some starter, to keep the amount of flour used for feedings modest : imagine at one stage you have 4 tablespoons of starter and you feed it 2 tablespoons each of flour and water. By the next feeding, you will have 8 tablespoons of starter, which you would have to feed 4 tablespoons each of flour and water, and then you would have 16 tablespoons of starter by the next feeding, which you will have to feed 8 tablespoons each of flour and water, etc ... so although it might seem wasteful to discard some of the starter (usually half), in the long run you end up using less flour for feedings. We just keep the quantities small by starting with a smaller amount of flour to start with.
What happens when you are creating your sourdough starter is that you are growing the yeasts and bacteria naturally present on the flour and in the environment (the air in your kitchen), so ideally you really want to use organic, stoneground, non enriched and certainly not bleached flour, for best results. You could use organic stoneground white, wholemeal, spelt, or rye flours. You will also get better results (better rise in particular) if you use strong flour, also called bread flour, which has a higher gluten content.
Also, it is best to use filtered water, or water that has been boiled and allowed to cool in an open vessel, as the chlorine in water could negatively impact your results.