1) craft some empty chests in your new home
2) leave everything but the very basics behind while you schlep to your old house
3) fill your inventory from most-needed stuff to least-needed stuff
4) schlep back to your new place
5) repeat from step 2 until your old chests are empty or you've had enough of moving stuff
You can also dismantle and move your empty chests, since chests are stackable (explain the logic to me. A steamer trunk can stack, but a pieces of ham steak can't?) you could plausibly cart a whole bunch of them around.
TIP: for the trek to your new base, you can flick the difficulty to peaceful. Some might consider it cheating, but it makes things go more smoothly. Or you can wait until night on hard difficulty to card your diamonds, gold and iron.
The Meaning of Life, the Universe, and Everything.
I did it very successfully with a variant of what internutter was saying. Obviously, you'll have storage in the old place, and empty storage of equal size in the new place. Since I did want to keep using all chests, and I wanted to cut the length of the trip down, I also made a storage house halfway between them. I was by the shore, which really helps cut down on beasties, but I did it all during the day, so it wasn't much of a factor. So in a few trips, I moved things from the old place, to the storage house, and then to the new place, and then I used the storage place chests for other things.
It's a process. Carts or boats help speed it up. The more houses/forts you have, the more secure you are from creeper assault loss.