To ‘get a foot in the door’ ‘To get my/your/… foot in the door’
means to get part of the way in. Think of entering a house, you usually put your whole body in, not just a foot. If you wished to work for a company, although you might desire a high position, you might be pleased to get started at a low position, just like getting your foot in the door.
To put your foot in your mouth. (the phrase is not in someone else’s mouth.) it means to make a mistake, saying something inappropriate. Think of putting your foot in the right place, not in the wrong place. Your mouth is not for that!