Every move is unique as far as budgeting goes, but there are some basics that are important to remember in your budget.
Deposit(s)- If you are going to rent an apartment or a house you will need a deposit. This is usually about $500 for an apartment or equal to your monthly rent if you are renting a house. You will also be expected to give the first month's rent, and occasionally the last month's rent as well. You will also need deposit money for a storage unit if you plan on renting one.
Transportation - Weather you drive or fly, you need to get to your new home somehow. If you are driving don't forget to budget food and snacks for the road trip. If you are going more the 5 hours it is a good idea to budget enough money for a hotel in case you get tired or road conditions aren't great.
Hotel- You may need to stay in a hotel while your things are in transit or waiting for your new place to become available.
Moving Company - If you are moving yourself, or hiring someone to do it for you, it is best to call a couple of companies for quotes on boxes, truck, services, and any other things you will need for your move. Include mileage and est. how much weight you have, most companies have simple calculators for you to use.
Food- With all your cooking equipment packed, you will need to eat a lot of fast food and/or convenience foods. If you have a special diet you may need to budget more to accommodate your needs.
Bills- Just because your address changes doesn't mean you don't have to pay your bills on time. Make a couple of every bill that comes in the mail for a month and then you will know when, who, and how much.
That is just the beginning, as I said everybody's budget will be different. I just wanted to give you an idea on what kinds of things to remember to calculate into your moving budget.