all you really need to get started sewing--besides something to sew, like a tear in a stuffed animal, or a patch to put on a jacket--is a needle and some thread. if you don't already have any, you can ask your housemates, family, or friends if they have any you can use. you can also check thrift stores or flea markets to get a variety of thread colors for a low price.
of course it's also nice to have scissors--necessary if you're going to be cutting fabric, but you can get away with just nail clippers if you're putting together a pocket sewing kit. if you're seriously get into sewing, you may want to have one pair of sharp scissors that you use only for cutting fabric, since this can help keep them sharp.
a set of pins can help keep fabric in place while you're sewing. you'll probably also want somewhere easy to put them down so they don't get lost, and easy to pick them back up when you need them again. we use a small box, some other options are a magnet or a pin cushion.
a measuring tape or ruler can be useful to make sure you're cutting something to the right size, and a pencil or chalk can be used to mark where you want to cut or sew. if you plan on altering clothes that are already made, a seam ripper makes removing lots of stitches at once much quicker than pulling them out by hand.
it can also be nice to make a spot to keep things like buttons and fabric scraps from your worn-out clothes, so you can use them in future repairs.