No one will be able to answer your question because we do not know your zone, climate, weather or area. Also, we do not know what you are wanting to transplant, size of the plant, etc.

So......if you provide a bit more info, maybe someone here can help you.

I live in Central/South New Jersey, and it's Spring time here now w/ moderately warm weather. I want to redo the landscaping in my back yard and move/re-plant some of the bushes (unknown), and trees (one is a fruit tree, other is a holly tree), but I was told that it's too late in the season to dig and then re-plant them, so I have to wait until the Fall. I just want to find out if that's true.

Impossible to give advice without knowing what kind of bushes you have,,,,
How can you know where to put them without knowing what they are???

It's not impossible to answer this question. You CAN move them now, but I'd recommend against it. First, it is peak growing season for most plants in the northern hemisphere and transplanting/moving established trees and shrubs will put them in shock if they are actively growing. Second, we are approaching the time of year that tends to be the hottest and driest in most parts of the country and newly planted material is difficult to maintain in good health without very careful attention, specially with regards to irrigation. The combination of inevitable transplant shock and the heat and dryness of summer just do not make for very successful transplanting now, even for very experienced gardeners.

Fall is a much better choice - the plants approach dormancy when the weather cools and winter nears, resulting in greatly reduced risk of transplant shock and the weather is typically more conducive to this process, with rain more frequent. Mid to late September will still allow plenty of time for plants to settle in before winter cold and bad weather.

Maybe if you take photos of your plants and post them here or on name that plant forum you will have time over the summer to ID them and then decide which ones will be happiest and fit your needs best in what part of your yard. Not all plants will be equally successful in all conditions.