If I manage to obtain an I-20, does that automatically invalidate my U.S. tourist visa? Or do the visa status and the I-20 exist independently of each other?
Visa and status are different things — it’s important to understand that right away. A visa gives you the right to cross the border and enter the country, while status determines your position once you’re inside. Obtaining an I-20 and getting F-1 status doesn’t automatically cancel your tourist visa — it just becomes irrelevant while you’re in the country. If you decide to leave and return, and your tourist visa has expired by then, you’ll have to apply for a student visa again through the embassy.
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Got it, thanks for the clarification. So as long as I’m in the country on student status there aren’t any problems, but when I leave I’ll have to start thinking about a new visa right away.
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One nuance to know — a tourist visa may not be physically expired, but after switching to student status it still won’t help when returning to the US. You have to enter with intent that matches your status, and the border officer will see you’re coming to study — so you need an F-1 visa. In practice, after changing status you become essentially unable to leave and re-enter until you obtain an F-1 through a consulate.
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Don’t panic yet — first calmly get your I-20 and change of status sorted, and you can think about leaving later. I went through the same thing: at first it all seemed confusing, but then one thing gets done after another. The main thing is to keep copies of your documents and don’t rush 
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From experience — in the first year after changing your status, don’t plan to travel; give yourself time to settle in and get used to it. Later, when you need to go home, you can calmly apply for a student visa at the embassy without rushing. Keep copies of your I-20 and SEVIS handy )
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