What are Some Common Extra Fees and Costs of Moving to a New Apartment? As you get ready to move to a new apartment, you probably are budgeting for some of the expenses that the move will ask for. This can include some items like the first month’s rent and a security deposit on the new apartment. The landlord will be able to discuss these amounts with you when you do the tour or sometime before you move into the unit. But those are just a few of the fees that...