Yup, I've made a buncha various cheese with storeboughten milk -- the caveat being, this is from a local dairy co-op, and is both pretty fresh AND not pasteurized all that hot. Having experimented around (and wasted a certain amount

) I would say that there are some things you can do and some things you can't.
If you have to use national-brand American grocery store milk, I would say you may be limited to paneer/"whole milk ricotta"/acid-coagulated type things, from which a version of ricotta salata can also be made. You could TRY doing a basic farmer cheese type hard cheese, with added calcium chloride, and see what happens. But with milk worse than what I'm working with from my store, which is already fairly limiting, I don't think you'd expect to do much with older hotter-pasteurized stuff.
Yes, that's why I got sheep
I have not unfortunately really got them into production-type milking this year, due to one thing and another, and now it is almost too late in their lactation to do much good. If circumstances allow me to still have them next year though I will be FAR better prepared and hopefully WILL make a significant amount of sheep yogurt and sheep cheese.
But, back to your situation -- just get the freshest and coolest-pasteurized milk you can, and try some simple things and see what happens.
Good luck, have fun,
Pat