I would like to transplant my front-of-the-house shrubs and plants to my backyard because I don't like them very much. I then want to put in new shrubs/plants in front of my house. They will go on both sides of my front steps, 3-5 plants/shrubs on each side. I thought that fall was known as the best time of the year to do this, but I am worried that they won't have enough time to really settle in and grow in roots for the harsh New England winters. Is it better to do this in fall or spring?
It's better to do it in fall, but before the first frost date, whenever that is in your USDA zone. It's not a good idea to plant small cuttings or seedlings at that time, but containerised or wrapped, bare root plants of a reasonable size will be fine. https://newengland.com/today/living/gardening/fall-planting/
The fall IS the best time to transplant. It is also the best time to purchase new plants! They are half price or better. If you want to transplant your old plants from the front to the back, this is the only time to do so. How established are your front yard plants? If you really want to make sure they survive the transplanting there are steps to take and possibly an extra year.
Have you purchased your new plants? What ideas have you got for different plants in the front. What is it you don't like about the plantings in your front yard? What exactly are you wanting to move to the backyard from the front yard? Do not fertilize those front yard plants at all if they will be going through a major stress of transplanting and this can only be done in the late fall before the ground freezes.
How long ago were the plants you want to transplant planted in your front yard? How large and what species are they?