Photoshop to illustrator for shirt design :(

i have been using photoshop for years but i have never touched illustrator. a friend that i had designed for wanted a couple of those designs made into t-shirts. the problem is, the designs aren’t vectors but the guy at the printshop said that i would be able to convert them to ai format fairly easily.
i got a trial of illustrator and tried my best but to me it doesn’t look right.
is it possible to turn a raster image into a vector?
im totally lost here.:pa: