The first step in removing gum from laundry is to harden it by using an ice cube or put the garment in a plastic bag in the freezer for several hours. Take it out of the freezer and while the gum is hard, scrape it off with a dull knife or straight edge. If it doesn't all come off, refreeze and try again.
When all of the gum has been removed, if the garment is washable treat the remaining spot with an equal mixture of liquid dishwashing soap and white distilled vinegar. Rub in gently and then wash the garment as usual. Check the stain before drying, if it remains, treat again.
If the gum is old and dry, use some petroleum jelly to loosen the gum. Use just a dab and work it in well. Then you must treat the fabric to remove the grease!
If the garment is dry clean only, use the freezing technique to remove the gum then point out and identify the stain to your professional cleaner. If you are using a home dry cleaning kit, be sure to treat the stain with the provided stain remover before putting the garment in the dryer bag.
Specific questions? Just ask here.