Illinois is bitterly cold in the winter (mid-November to mid-March in theory, though it's stayed here all the way to end-March and counting this year). Just about nothing that you can buy in India will help you here, so you will need to buy a thick winter jacket, preferably of down feather, as well as snow boots (this might not be necessary; I just wear sneakers) later on.
However, when you arrive, it will be summer and that is extremely hot. But since the state is very windy courtesy of Lake Michigan, you will need to carry a light jacket for the evenings initially (wear it on the flight). Therefore, carry a mix of light clothing and thick clothing for the different seasons; do carry thermals and a towel or two. You can carry shorts or cargoes and wear them to class - nobody cares. T-shirts and casual shirts are fine; I do recommend you to carry a suit (folded in whatever way possible) because that can be expensive here and you will definitely need it some time or the other. Gloves are better purchased here; the woolen ones used in India will be useless when it snows (which is at least 25% of the year).
Finally, remember that clothes can always be purchased here and they are not too expensive, provided you buy them from the right place. So, don't stuff your suitcase in panic. Most homes have a laundry room to wash clothes, but some have ones right in the apartment (that's the best thing you can get). You will need to buy some liquid detergent (again, Walmart); laundry rooms would be coin-operated, so you will need to get some coins at coin vending machines (more on money later). Typically, people wash clothes once a week, although I do it once in two weeks; it depends entirely on you. Here's a neat tip: clothes come out piping hot from the dryer and you can simply pat the wrinkles out before they cool; this eliminates the need to iron clothes. Of course, formal shirts still need to be ironed, but that's only once in a while. You will probably never have to iron your daily clothes. An electric iron will cost less than $10.
6. Food & Cutlery
If you can't cook, you will either go broke or die of hunger. So, learn to cook. You can buy all cutlery (spoons, forks, butter knives, pots, plates, pans etc.) from here, but I do recommend you get a pressure cooker from home (check if you have a gas or electric stove because the cooker type changes; default is electric), because that can be a little hard to get in the US as Americans don't use it too much, while Indians use it a lot. All other cutlery are optional - you might like to get a favorite spoon or something. Your apartment will most probably have a microwave, but you will need to purchase a toaster ($8 if I remember right; a microwave is for $20, a blender $12). As you can see, these basic things are pretty cheap here and with $100 you can setup an entire family! You will also need to buy a sponge and dishwashing liquid; you can use a dishwasher, but that will spike up your electricity bill and honestly, you won't have enough dishes to warrant its need.
But coming to more specific things: vegetables can be purchased here. Except big stuff like gobi or aloo, I usually buy everything in cans (vegetables, corn, tomato sauce etc.) because they are just so much more convenient, albeit unhealthy (but come to think of it, given how clean the air is here compared to India, it's a zero sum trade-off). From India, I recommend you get some of those ready-to-eat things to last for the first few days while you settle down. I also got myself some Maggi because, well, I love maggi :D All of that can also be purchased at an Indian store, although it is expensive (for Maggi you can buy Ramen noodles, which are cheap).
It is not all that hard being a vegetarian here - you have plenty of options, you just need to ask. They can usually create a vegetarian version of anything as well. But, if you can't cook, you will eventually run out of options. So, learn to cook.
Finally, you should get three basic masalas from India: kitchen king, lal mirch and haldi. Almost all cooking can be done with just these three (you can get salt here), I'm sure your mom will corroborate that. Other optional ones are rajma and chole masala. Again, all this is available at an Indian store, but it is expensive. For the first month, before you get paid, I think you should try to work with stuff from home. And learn to love cooking - you have absolutely no choice.