There are provisions at your state MVA for gifting a vehicle to an immediate family member - who would then owe no sales tax for the gift. It costs $20-25 here & may be different in your state.
Upon a change of ownership/title, the new owner buys his/her own insurance - since you are no longer the owner.
You cannot give a car to someone without signing over the title. Most states also require you to remove the license plates at the time of the sale/family gift and turn them in before you cancel your insurance.
That is how it works for a car with no loan. If you were planning to continue to pay the loan on someone else's behalf, you should contact the bank and notify them of the change - or better yet, contact them before you sign over the title.