I've noticed that in jeans, especially new ones, the colour tends to leak from them. Are there any ways to mitigate this? I've heard of washing them in cold water, turning them inside out, and adding a cup of white vinegar. Should these be done each time to keep the original color intact?
New denim clothing they may have "loose dye" in them, so yeah wash them several times in cold water to wash the loose dye out, from then on you can wash them with your other clothes, but I'd recommend washing them with dark colours and turning them inside out stops them rubbing against the other clothes and rubbing off the dye. That's all I can suggest.
If you're convinced the vinegar rinse works, yes, you're supposed to use it every time, in the last rinse, and it has to be white vinegar. I never found it made any difference though - if your jeans are indigo dyed, then they 'bleed' every time you wash them, and when they're new, the colour bleeds or transfers onto pale furniture and carpets if you sit on them.
Definitely turn them inside out every time before washing, always do up the buttons and zipper, use cold water for the whole wash unless they're particularly dirty, do use a liquid wash intended for dark clothing, don't dry them in direct sunlight - this helps to keep the colour for longer. If your jeans aren't fashionably 'patchy', with paler areas for aesthetic effect, you can buy an indigo jeans dye and re dye them when they get particularly faded.
The results from using this aren't quite the same as brand new jeans though.