Assuming bed-size means twin, you should be find to pop it in the washer. I use a dye-free detergent like Tide Free or All Free. I never use anything else (NO Clorox 2 or fabric softener). I wash twice in the coldest water setting on the slowest speed for the shortest cycle (Tap Water Cold / Handwash Cycle / 6 minute agitation). Then I put it in the dryer on Low heat until it is mostly dry. I hang it over the shower curtain rod or lay it flat to dry (clothes pins distort the quilt too much for me).
I do this for all my quilts (100% cotton fabric, thread & batting -- though I've also done a poly batt for home dec items) and they all come out great! No bleeding. No shrinking. Just that nice crinkly look that all cotton quilts get when they are loved. I like washing & drying twice because it makes the quilt extra soft to present to the recipient.