You can. Supermarket milk is absolutely fine BUT you need to add about 1/4 t of calcium chloride (food grade - not what you might use to melt ice), to each gallon of milk. That said, Mozzarella cheese is not really an easy cheese to make. It stretches only when the pH of the curds are between 5.2 and 5.5. Too much or too little acidity and it doesn't stretch and too much handling can make it grainy.
What I would do before trying to make just about any cheese is to get hold of any basic book on cheese making. Almost all of them (perhaps all of them) are organized from easier to harder cheeses to make so that they scaffold the skills you need. The easiest cheeses are fresh cheeses which tend to be soft and by definition don't need aging. To make harder cheeses you will need a rig of some kind to press the cheese under weight. Some cheeses need little pressure to remove the whey. Others require 50 lbs or more of pressure. Simple cheeses like quark, paneer, haloumi or ricotta (which you can drain and make far more dry than you are likely to buy) and cheese like squeaky curd (as used in Poutine) or a feta -type cheese are all much easier to make for a beginner than Mozz. And if you can find non organic goats milk (organic is usually heated to a point that the calcium chloride cannot repair the damage to the calcium compounds and so the cheese cannot coagulate as needed) you can make a delightful chevre using the same recipe as quark.
One last thought. I make cheese from a gallon of store bought milk almost every week, and I don't but any basic cultures to acidify and flavor the milk. What I do is I make milk kefir from kefir grains and I use the kefir I make to culture the milk. Kefir contains just about every bacterium that you might buy to make meso- or thermophillic cheeses (cheddar to swiss-type to parmesan). For blue cheese you need the specific culture and although I have yet to try making a brie but I suspect that the kefir has the cultures you would buy to make brie.
