Less than you'd think. The trick is to start with a fresh cheese like paneer or ricotta. Hard cheeses need aging time and more of a commitment, so save those for later.
For paneer (a good first cheese), you need whole milk, yogurt, lemon juice, salt, a colander and cheesecloth. All of it is available at a regular supermarket.
For mozzarella (a good second step), you need:
- Whole or 2% milk, ideally not ultra-pasteurized, which often won't curdle properly
- Citric acid, a rennet tablet, and kosher salt
- Distilled (or dechlorinated) water
- A non-reactive pot (stainless steel or copper, not aluminum or cast iron)
- A kitchen thermometer, cheesecloth and a large glass bowl
If you don't want to track down ingredients one by one, a mozzarella or mixed cheese-making kit is usually cheaper than buying everything separately. After mozzarella, a fresh goat cheese like chèvre is a natural next step.
Related: What is whey and what can I do with it? · How do you make ricotta from whey? · What is paneer, and is it the same as ricotta? · What is the difference between ricotta, mascarpone, cottage cheese and cream cheese?