Day 1: In a glass or stoneware jar or bowl, combine ¼ Cup water with ⅓ Cup flour. Stir with a wooden spoon until the flour is totally mixed into the water. Cover with a loose-fitting lid, towel, or doubled cheesecloth.
Day 2: Uncover the starter and remove half of the contents, discard what you remove. Add ⅓ cup of flour and ¼ cup water to the remaining contents of the starter & stir until well combined. Cover and store the starter on the counter.
Days 3-7: Repeat the steps from day 2. You should notice that your starter almost doubles in size every 12-24 hours.
Day 8: The starter should be mature enough to attempt baking bread. It should peak 8-24 hours after feeding, have lots of bubbles, and smell like sourdough.
Notes
Temperature matters.
A sourdough starter will peak faster at higher temperatures, so store your starter in a warm place - but not too warm or it will kill the bacteria and yeast - no warmer than 110F. In the winter, I use a proofing box set at about 80F when I need my starter to peak faster. In the warmer months, it peaks quickly on the kitchen counter.
Keep bugs out.
Use a rubber band or piece of twine to fasten the cloth around the neck of the jar you use to store your sourdough starter. This will prevent fruit flies from getting into the starter.
Test the starter.
Perform the float test to see if your sourdough starter is ready to use. Just drop a spoonful of sourdough starter into a cup of water. If it floats, it is ready to use.
Less can be more.
I have found that I have the best result when I keep only as much starter as I need. When I am ready to bake, I use almost all of the active starter so that all that's left in the jar is a very small amount (called the "scrapings"). I feed this tiny amount of starter and it peaks up perfectly.
Hydration matters.
Certain recipes will call for higher or lower hydration starter. The level of hydration of your starter just refers to how much water is in it. A lower hydration starter has less water, so it is thicker. A higher hydration starter has more water, so it is thinner. I prefer a higher hydration, thin starter for making pizza crust. But I prefer a lower hydration, thicker starter for making artisan bread.
Don't throw it away.
Sourdough starters are incredibly resilient and adaptable. They are almost impossible to kill. If your starter isn't performing well, some adjustments may need to be made to the feeding schedule, type of flour or water, ratio of flour and water, storage temperature, etc.
But don't throw it away and start over! Just work with what you have. The only time I would throw a sourdough starter away is if it was black and moldy from top to bottom. Even if the top of the starter looks scary or a few bugs get in, just pour as much off the top as you can, transfer a tiny bit from the bottom into a clean jar, feed it, and proceed as usual.
I have been keeping a starter for over 6 years and have never had to start over. I even accidentally baked my starter jar once! Luckily, there was a tiny little bit of the middle that was still liquidy. I salvaged that little bit, fed it, and it was good as new!