So, this is more of a "I am moving across the country - what do?" type of thing.
I would not put anything into storage. Take it with you, sell it, donate it, or throw it away.
If you are relocating, then yes, you do need to bring all of your official documents. If you already bought your study materials, I would bring them with you. If they are textbooks you don't need - sell them.
As for your furniture and household objects - do a cost analysis.
For furniture, unless you have all antique/heirloom things, I would just sell what you have before you leave. Craigslist/yard sale it all away. You can buy new furniture pretty cheap. Taking a wad of cash across country is infinitely cheaper than lugging a uhaul.
Perishables - give away/donate/throw away/sell if you can. You can try to sell anything that is not opened. Anything opened you will be giving away. Do not buy more of anything perishable unless you need to. Use what you have.
If you have VERY nice kitchen gear (all-clad pots, cast iron skillets, oven safe dishes), I would take those with you. If it is stuff you bought at ikea or the equivalent - sell it.
Anything that has sentimental value - bring it with you.
Anything that is expensive electronic - bring it with you.
Essentials for moving into a new apartment(things to have on your person, not shipped)
sleeping bag/air mattress. (sheets if you are not bringing a sleeping bag) Pillow. Phone/Computer and their chargers. Clothes for two weeks. Two bath towels. One roll of toilet paper. Your identification documents (ssn card, birth certificate, passport). At least $200.00 in cash and your credit card(s).
Everything else you can buy when you are there (most immediately -shower curtain/hooks, tooth brush, tooth paste, floss, more toilet paper), but I would look up where the closest drug store to you is so that you don't have to struggle around with it on your first day.
Everything else that you decided not to ship (furniture, kitchen supplies, decorations) you can purchase online, from ikea, or via Craigslist.
good luck!