You can always do some online research to compare living costs (rent) in a specific area. As a general rule, the closer to the city = more costly.
Are you seeking an apartment, or single house? Do you have pets, if so then it'd be a criteria to look for pet-friendly rentals, which are usually harder to find, depending on where you're looking. Every state is different & has their own set of rules & costs.
Another thing to consider is that apartments provide all appliances, while some houses don't. Yard upkeep is usually the tenant's responsibility when renting a house, so if you're not into outdoor upkeep, then apartments might be for you.
I'm in California & as an example, the expense for me to move would be:
First & last month's rent
Cleaning deposit (close to the amount of one month's rent)
Pet deposit (about half of one month's rent)
Utility hookup (some require a deposit for first time customers)
Cable installment
Cost in gas for moving everything
Stocking the fridge & cupboards
Where I live & due to my choices, it'd cost me about $7K to move (give or take a few hundred). But, I'm picky & my standards are high. I prefer suburban residential single homes in a good location. For that, it'd cost more.
Hope that helps some
Shangrala