Easiest way without cheats or mods (or a complicated process):
Make a list of the items
Sell everything you want 'moved'
Add that amount to the simoleons to transfer
Buy it all back in the new house
Note: you can't buy items in the new house if the Sim(s) don't have the appropriate level in career for unique items (or the quests etc... to get unique items for the household). Only way to get around that is to go to the empty new house and add the unique items first before the Sim(s) buy the lot and move in.
The value of the unique items will be added to the cost of the lot, so, it's not cheating. ;)
