The State of Idaho makes it easy for you. It's actually attached to the bottom of the Title. If/when you sell the car (if you're the title holder, and not the bank), you simply fill in and detach the notice, pay something like $3 (it's very much worth it), and send it in to the state. It releases your liability as of the date of sale (not the date they receive it).
New Mexico has an online form you can fill out. Also, take the plates, unless, for some reason, your state requires the plates to stay with the car (IIRC, California is like that, or was, for several years).
I just trade in cars nowadays. I buy them new, and trade them in, usually for about $500, because I ride ‘em hard and put ‘em away wet (VERY high mileage and almost certainly in need of major repairs).
I hate selling used cars to private parties.