When should I sow my broccoli seeds indoors and when should I transplant outside?
Look up the last frost date for your zip code here. Duluth, for example, has a last frost date of May 21.
Or, as Ed Staub points out in a comment below, for a better picture of the situation go to the original source of the data at the National Climatic Data Center (1971-2000):
You can transplant broccoli outside a week or two before the last frost date, and start the seed 4 to 6 weeks prior to the transplant date.
About a week before you transplant, start hardening off the plants: first set them outside on a moderate day for a couple of hours, avoiding winds or too much intense sunlight. Then set them outside for longer periods each day. This lets them acclimate to outside conditions so they don't get a sudden shock.