My favorite style of door weather strip is the push-in kerf style. It stays in place and you don’t have to worry about it coming off of the door or jamb when the adhesive wears out.
Not all doors can use the kerf-style weather-stripping though. A kerf must be present in order for this weather strip to work. A kerf is a small groove that sits around the perimeter of the door. Look at the stop (the piece of wood that the exterior side of the door rests against). If you see a groove about 1/8” wide, your door can use this type of weather strip. If your door does have a kerf, you probably already have this weather strip in though. I sure wouldn’t replace it with the peel-and-stick style.
To remove the old kerf-style weather strip, just pull it out of the groove. There is a small barb on it that pushes into the kerf and holds. Measure the distance for the new piece and cut it. Start at one end and push it all the way into the kerf. For the corners, you can either cut the strip straight across (you may have some overlap in the corners), or you can cut the strip at 45-degree angles to make a 90-degree corner (for example, like you would on window trim). You shouldn’t have any problems with it coming out of the groove.